1.1 --- a/src/HOL/Algebra/QuotRing.thy Mon Sep 19 22:48:05 2011 +0200
1.2 +++ b/src/HOL/Algebra/QuotRing.thy Mon Sep 19 23:18:18 2011 +0200
1.3 @@ -10,8 +10,7 @@
1.4
1.5 subsection {* Multiplication on Cosets *}
1.6
1.7 -definition
1.8 - rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"
1.9 +definition rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"
1.10 ("[mod _:] _ \<Otimes>\<index> _" [81,81,81] 80)
1.11 where "rcoset_mult R I A B = (\<Union>a\<in>A. \<Union>b\<in>B. I +>\<^bsub>R\<^esub> (a \<otimes>\<^bsub>R\<^esub> b))"
1.12
1.13 @@ -19,86 +18,71 @@
1.14 text {* @{const "rcoset_mult"} fulfils the properties required by
1.15 congruences *}
1.16 lemma (in ideal) rcoset_mult_add:
1.17 - "\<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
1.18 -apply rule
1.19 -apply (rule, simp add: rcoset_mult_def, clarsimp)
1.20 -defer 1
1.21 -apply (rule, simp add: rcoset_mult_def)
1.22 -defer 1
1.23 + "x \<in> carrier R \<Longrightarrow> y \<in> carrier R \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
1.24 + apply rule
1.25 + apply (rule, simp add: rcoset_mult_def, clarsimp)
1.26 + defer 1
1.27 + apply (rule, simp add: rcoset_mult_def)
1.28 + defer 1
1.29 proof -
1.30 fix z x' y'
1.31 assume carr: "x \<in> carrier R" "y \<in> carrier R"
1.32 - and x'rcos: "x' \<in> I +> x"
1.33 - and y'rcos: "y' \<in> I +> y"
1.34 - and zrcos: "z \<in> I +> x' \<otimes> y'"
1.35 + and x'rcos: "x' \<in> I +> x"
1.36 + and y'rcos: "y' \<in> I +> y"
1.37 + and zrcos: "z \<in> I +> x' \<otimes> y'"
1.38
1.39 - from x'rcos
1.40 - have "\<exists>h\<in>I. x' = h \<oplus> x" by (simp add: a_r_coset_def r_coset_def)
1.41 - from this obtain hx
1.42 - where hxI: "hx \<in> I"
1.43 - and x': "x' = hx \<oplus> x"
1.44 - by fast+
1.45 -
1.46 - from y'rcos
1.47 - have "\<exists>h\<in>I. y' = h \<oplus> y" by (simp add: a_r_coset_def r_coset_def)
1.48 - from this
1.49 - obtain hy
1.50 - where hyI: "hy \<in> I"
1.51 - and y': "y' = hy \<oplus> y"
1.52 - by fast+
1.53 + from x'rcos have "\<exists>h\<in>I. x' = h \<oplus> x"
1.54 + by (simp add: a_r_coset_def r_coset_def)
1.55 + then obtain hx where hxI: "hx \<in> I" and x': "x' = hx \<oplus> x"
1.56 + by fast+
1.57
1.58 - from zrcos
1.59 - have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')" by (simp add: a_r_coset_def r_coset_def)
1.60 - from this
1.61 - obtain hz
1.62 - where hzI: "hz \<in> I"
1.63 - and z: "z = hz \<oplus> (x' \<otimes> y')"
1.64 - by fast+
1.65 + from y'rcos have "\<exists>h\<in>I. y' = h \<oplus> y"
1.66 + by (simp add: a_r_coset_def r_coset_def)
1.67 + then obtain hy where hyI: "hy \<in> I" and y': "y' = hy \<oplus> y"
1.68 + by fast+
1.69 +
1.70 + from zrcos have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')"
1.71 + by (simp add: a_r_coset_def r_coset_def)
1.72 + then obtain hz where hzI: "hz \<in> I" and z: "z = hz \<oplus> (x' \<otimes> y')"
1.73 + by fast+
1.74
1.75 note carr = carr hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
1.76
1.77 from z have "z = hz \<oplus> (x' \<otimes> y')" .
1.78 - also from x' y'
1.79 - have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
1.80 - also from carr
1.81 - have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
1.82 - finally
1.83 - have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
1.84 + also from x' y' have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
1.85 + also from carr have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
1.86 + finally have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
1.87
1.88 - from hxI hyI hzI carr
1.89 - have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I" by (simp add: I_l_closed I_r_closed)
1.90 + from hxI hyI hzI carr have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"
1.91 + by (simp add: I_l_closed I_r_closed)
1.92
1.93 - from this and z2
1.94 - have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
1.95 - thus "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
1.96 + with z2 have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
1.97 + then show "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
1.98 next
1.99 fix z
1.100 assume xcarr: "x \<in> carrier R"
1.101 - and ycarr: "y \<in> carrier R"
1.102 - and zrcos: "z \<in> I +> x \<otimes> y"
1.103 - from xcarr
1.104 - have xself: "x \<in> I +> x" by (intro a_rcos_self)
1.105 - from ycarr
1.106 - have yself: "y \<in> I +> y" by (intro a_rcos_self)
1.107 -
1.108 - from xself and yself and zrcos
1.109 - show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" by fast
1.110 + and ycarr: "y \<in> carrier R"
1.111 + and zrcos: "z \<in> I +> x \<otimes> y"
1.112 + from xcarr have xself: "x \<in> I +> x" by (intro a_rcos_self)
1.113 + from ycarr have yself: "y \<in> I +> y" by (intro a_rcos_self)
1.114 + show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b"
1.115 + using xself and yself and zrcos by fast
1.116 qed
1.117
1.118
1.119 subsection {* Quotient Ring Definition *}
1.120
1.121 -definition
1.122 - FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring" (infixl "Quot" 65)
1.123 +definition FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"
1.124 + (infixl "Quot" 65)
1.125 where "FactRing R I =
1.126 - \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I, one = (I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>), zero = I, add = set_add R\<rparr>"
1.127 + \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I,
1.128 + one = (I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>), zero = I, add = set_add R\<rparr>"
1.129
1.130
1.131 subsection {* Factorization over General Ideals *}
1.132
1.133 text {* The quotient is a ring *}
1.134 -lemma (in ideal) quotient_is_ring:
1.135 - shows "ring (R Quot I)"
1.136 +lemma (in ideal) quotient_is_ring: "ring (R Quot I)"
1.137 apply (rule ringI)
1.138 --{* abelian group *}
1.139 apply (rule comm_group_abelian_groupI)
1.140 @@ -112,15 +96,15 @@
1.141 apply (clarify)
1.142 apply (simp add: rcoset_mult_add, fast)
1.143 --{* mult @{text one_closed} *}
1.144 - apply (force intro: one_closed)
1.145 + apply force
1.146 --{* mult assoc *}
1.147 apply clarify
1.148 apply (simp add: rcoset_mult_add m_assoc)
1.149 --{* mult one *}
1.150 apply clarify
1.151 - apply (simp add: rcoset_mult_add l_one)
1.152 + apply (simp add: rcoset_mult_add)
1.153 apply clarify
1.154 - apply (simp add: rcoset_mult_add r_one)
1.155 + apply (simp add: rcoset_mult_add)
1.156 --{* distr *}
1.157 apply clarify
1.158 apply (simp add: rcoset_mult_add a_rcos_sum l_distr)
1.159 @@ -131,8 +115,7 @@
1.160
1.161 text {* This is a ring homomorphism *}
1.162
1.163 -lemma (in ideal) rcos_ring_hom:
1.164 - "(op +> I) \<in> ring_hom R (R Quot I)"
1.165 +lemma (in ideal) rcos_ring_hom: "(op +> I) \<in> ring_hom R (R Quot I)"
1.166 apply (rule ring_hom_memI)
1.167 apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
1.168 apply (simp add: FactRing_def rcoset_mult_add)
1.169 @@ -140,8 +123,7 @@
1.170 apply (simp add: FactRing_def)
1.171 done
1.172
1.173 -lemma (in ideal) rcos_ring_hom_ring:
1.174 - "ring_hom_ring R (R Quot I) (op +> I)"
1.175 +lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) (op +> I)"
1.176 apply (rule ring_hom_ringI)
1.177 apply (rule is_ring, rule quotient_is_ring)
1.178 apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
1.179 @@ -156,13 +138,14 @@
1.180 shows "cring (R Quot I)"
1.181 proof -
1.182 interpret cring R by fact
1.183 - show ?thesis apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
1.184 - apply (rule quotient_is_ring)
1.185 - apply (rule ring.axioms[OF quotient_is_ring])
1.186 -apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric])
1.187 -apply clarify
1.188 -apply (simp add: rcoset_mult_add m_comm)
1.189 -done
1.190 + show ?thesis
1.191 + apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
1.192 + apply (rule quotient_is_ring)
1.193 + apply (rule ring.axioms[OF quotient_is_ring])
1.194 + apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric])
1.195 + apply clarify
1.196 + apply (simp add: rcoset_mult_add m_comm)
1.197 + done
1.198 qed
1.199
1.200 text {* Cosets as a ring homomorphism on crings *}
1.201 @@ -171,65 +154,57 @@
1.202 shows "ring_hom_cring R (R Quot I) (op +> I)"
1.203 proof -
1.204 interpret cring R by fact
1.205 - show ?thesis apply (rule ring_hom_cringI)
1.206 - apply (rule rcos_ring_hom_ring)
1.207 - apply (rule is_cring)
1.208 -apply (rule quotient_is_cring)
1.209 -apply (rule is_cring)
1.210 -done
1.211 + show ?thesis
1.212 + apply (rule ring_hom_cringI)
1.213 + apply (rule rcos_ring_hom_ring)
1.214 + apply (rule is_cring)
1.215 + apply (rule quotient_is_cring)
1.216 + apply (rule is_cring)
1.217 + done
1.218 qed
1.219
1.220
1.221 subsection {* Factorization over Prime Ideals *}
1.222
1.223 text {* The quotient ring generated by a prime ideal is a domain *}
1.224 -lemma (in primeideal) quotient_is_domain:
1.225 - shows "domain (R Quot I)"
1.226 -apply (rule domain.intro)
1.227 - apply (rule quotient_is_cring, rule is_cring)
1.228 -apply (rule domain_axioms.intro)
1.229 - apply (simp add: FactRing_def) defer 1
1.230 - apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify)
1.231 - apply (simp add: rcoset_mult_add) defer 1
1.232 +lemma (in primeideal) quotient_is_domain: "domain (R Quot I)"
1.233 + apply (rule domain.intro)
1.234 + apply (rule quotient_is_cring, rule is_cring)
1.235 + apply (rule domain_axioms.intro)
1.236 + apply (simp add: FactRing_def) defer 1
1.237 + apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify)
1.238 + apply (simp add: rcoset_mult_add) defer 1
1.239 proof (rule ccontr, clarsimp)
1.240 assume "I +> \<one> = I"
1.241 - hence "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
1.242 - hence "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
1.243 - from this and a_subset
1.244 - have "I = carrier R" by fast
1.245 - from this and I_notcarr
1.246 - show "False" by fast
1.247 + then have "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
1.248 + then have "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
1.249 + with a_subset have "I = carrier R" by fast
1.250 + with I_notcarr show False by fast
1.251 next
1.252 fix x y
1.253 assume carr: "x \<in> carrier R" "y \<in> carrier R"
1.254 - and a: "I +> x \<otimes> y = I"
1.255 - and b: "I +> y \<noteq> I"
1.256 + and a: "I +> x \<otimes> y = I"
1.257 + and b: "I +> y \<noteq> I"
1.258
1.259 have ynI: "y \<notin> I"
1.260 proof (rule ccontr, simp)
1.261 assume "y \<in> I"
1.262 - hence "I +> y = I" by (rule a_rcos_const)
1.263 - from this and b
1.264 - show "False" by simp
1.265 + then have "I +> y = I" by (rule a_rcos_const)
1.266 + with b show False by simp
1.267 qed
1.268
1.269 - from carr
1.270 - have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self)
1.271 - from this
1.272 - have xyI: "x \<otimes> y \<in> I" by (simp add: a)
1.273 + from carr have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self)
1.274 + then have xyI: "x \<otimes> y \<in> I" by (simp add: a)
1.275
1.276 - from xyI and carr
1.277 - have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
1.278 - from this and ynI
1.279 - have "x \<in> I" by fast
1.280 - thus "I +> x = I" by (rule a_rcos_const)
1.281 + from xyI and carr have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
1.282 + with ynI have "x \<in> I" by fast
1.283 + then show "I +> x = I" by (rule a_rcos_const)
1.284 qed
1.285
1.286 text {* Generating right cosets of a prime ideal is a homomorphism
1.287 on commutative rings *}
1.288 -lemma (in primeideal) rcos_ring_hom_cring:
1.289 - shows "ring_hom_cring R (R Quot I) (op +> I)"
1.290 -by (rule rcos_ring_hom_cring, rule is_cring)
1.291 +lemma (in primeideal) rcos_ring_hom_cring: "ring_hom_cring R (R Quot I) (op +> I)"
1.292 + by (rule rcos_ring_hom_cring) (rule is_cring)
1.293
1.294
1.295 subsection {* Factorization over Maximal Ideals *}
1.296 @@ -243,106 +218,92 @@
1.297 shows "field (R Quot I)"
1.298 proof -
1.299 interpret cring R by fact
1.300 - show ?thesis apply (intro cring.cring_fieldI2)
1.301 - apply (rule quotient_is_cring, rule is_cring)
1.302 - defer 1
1.303 - apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp)
1.304 - apply (simp add: rcoset_mult_add) defer 1
1.305 -proof (rule ccontr, simp)
1.306 - --{* Quotient is not empty *}
1.307 - assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
1.308 - hence II1: "I = I +> \<one>" by (simp add: FactRing_def)
1.309 - from a_rcos_self[OF one_closed]
1.310 - have "\<one> \<in> I" by (simp add: II1[symmetric])
1.311 - hence "I = carrier R" by (rule one_imp_carrier)
1.312 - from this and I_notcarr
1.313 - show "False" by simp
1.314 -next
1.315 - --{* Existence of Inverse *}
1.316 - fix a
1.317 - assume IanI: "I +> a \<noteq> I"
1.318 - and acarr: "a \<in> carrier R"
1.319 + show ?thesis
1.320 + apply (intro cring.cring_fieldI2)
1.321 + apply (rule quotient_is_cring, rule is_cring)
1.322 + defer 1
1.323 + apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp)
1.324 + apply (simp add: rcoset_mult_add) defer 1
1.325 + proof (rule ccontr, simp)
1.326 + --{* Quotient is not empty *}
1.327 + assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
1.328 + then have II1: "I = I +> \<one>" by (simp add: FactRing_def)
1.329 + from a_rcos_self[OF one_closed] have "\<one> \<in> I"
1.330 + by (simp add: II1[symmetric])
1.331 + then have "I = carrier R" by (rule one_imp_carrier)
1.332 + with I_notcarr show False by simp
1.333 + next
1.334 + --{* Existence of Inverse *}
1.335 + fix a
1.336 + assume IanI: "I +> a \<noteq> I" and acarr: "a \<in> carrier R"
1.337
1.338 - --{* Helper ideal @{text "J"} *}
1.339 - def J \<equiv> "(carrier R #> a) <+> I :: 'a set"
1.340 - have idealJ: "ideal J R"
1.341 - apply (unfold J_def, rule add_ideals)
1.342 - apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr)
1.343 - apply (rule is_ideal)
1.344 - done
1.345 + --{* Helper ideal @{text "J"} *}
1.346 + def J \<equiv> "(carrier R #> a) <+> I :: 'a set"
1.347 + have idealJ: "ideal J R"
1.348 + apply (unfold J_def, rule add_ideals)
1.349 + apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr)
1.350 + apply (rule is_ideal)
1.351 + done
1.352
1.353 - --{* Showing @{term "J"} not smaller than @{term "I"} *}
1.354 - have IinJ: "I \<subseteq> J"
1.355 - proof (rule, simp add: J_def r_coset_def set_add_defs)
1.356 - fix x
1.357 - assume xI: "x \<in> I"
1.358 - have Zcarr: "\<zero> \<in> carrier R" by fast
1.359 - from xI[THEN a_Hcarr] acarr
1.360 - have "x = \<zero> \<otimes> a \<oplus> x" by algebra
1.361 + --{* Showing @{term "J"} not smaller than @{term "I"} *}
1.362 + have IinJ: "I \<subseteq> J"
1.363 + proof (rule, simp add: J_def r_coset_def set_add_defs)
1.364 + fix x
1.365 + assume xI: "x \<in> I"
1.366 + have Zcarr: "\<zero> \<in> carrier R" by fast
1.367 + from xI[THEN a_Hcarr] acarr
1.368 + have "x = \<zero> \<otimes> a \<oplus> x" by algebra
1.369 + with Zcarr and xI show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
1.370 + qed
1.371
1.372 - from Zcarr and xI and this
1.373 - show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
1.374 + --{* Showing @{term "J \<noteq> I"} *}
1.375 + have anI: "a \<notin> I"
1.376 + proof (rule ccontr, simp)
1.377 + assume "a \<in> I"
1.378 + then have "I +> a = I" by (rule a_rcos_const)
1.379 + with IanI show False by simp
1.380 + qed
1.381 +
1.382 + have aJ: "a \<in> J"
1.383 + proof (simp add: J_def r_coset_def set_add_defs)
1.384 + from acarr
1.385 + have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
1.386 + with one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup]
1.387 + show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
1.388 + qed
1.389 +
1.390 + from aJ and anI have JnI: "J \<noteq> I" by fast
1.391 +
1.392 + --{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
1.393 + from idealJ and IinJ have "J = I \<or> J = carrier R"
1.394 + proof (rule I_maximal, unfold J_def)
1.395 + have "carrier R #> a \<subseteq> carrier R"
1.396 + using subset_refl acarr by (rule r_coset_subset_G)
1.397 + then show "carrier R #> a <+> I \<subseteq> carrier R"
1.398 + using a_subset by (rule set_add_closed)
1.399 + qed
1.400 +
1.401 + with JnI have Jcarr: "J = carrier R" by simp
1.402 +
1.403 + --{* Calculating an inverse for @{term "a"} *}
1.404 + from one_closed[folded Jcarr]
1.405 + have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
1.406 + by (simp add: J_def r_coset_def set_add_defs)
1.407 + then obtain r i where rcarr: "r \<in> carrier R"
1.408 + and iI: "i \<in> I" and one: "\<one> = r \<otimes> a \<oplus> i" by fast
1.409 + from one and rcarr and acarr and iI[THEN a_Hcarr]
1.410 + have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
1.411 +
1.412 + --{* Lifting to cosets *}
1.413 + from iI have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
1.414 + by (intro a_rcosI, simp, intro a_subset, simp)
1.415 + with rai1 have "a \<otimes> r \<in> I +> \<one>" by simp
1.416 + then have "I +> \<one> = I +> a \<otimes> r"
1.417 + by (rule a_repr_independence, simp) (rule a_subgroup)
1.418 +
1.419 + from rcarr and this[symmetric]
1.420 + show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
1.421 qed
1.422 -
1.423 - --{* Showing @{term "J \<noteq> I"} *}
1.424 - have anI: "a \<notin> I"
1.425 - proof (rule ccontr, simp)
1.426 - assume "a \<in> I"
1.427 - hence "I +> a = I" by (rule a_rcos_const)
1.428 - from this and IanI
1.429 - show "False" by simp
1.430 - qed
1.431 -
1.432 - have aJ: "a \<in> J"
1.433 - proof (simp add: J_def r_coset_def set_add_defs)
1.434 - from acarr
1.435 - have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
1.436 - from one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] and this
1.437 - show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
1.438 - qed
1.439 -
1.440 - from aJ and anI
1.441 - have JnI: "J \<noteq> I" by fast
1.442 -
1.443 - --{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
1.444 - from idealJ and IinJ
1.445 - have "J = I \<or> J = carrier R"
1.446 - proof (rule I_maximal, unfold J_def)
1.447 - have "carrier R #> a \<subseteq> carrier R"
1.448 - using subset_refl acarr
1.449 - by (rule r_coset_subset_G)
1.450 - from this and a_subset
1.451 - show "carrier R #> a <+> I \<subseteq> carrier R" by (rule set_add_closed)
1.452 - qed
1.453 -
1.454 - from this and JnI
1.455 - have Jcarr: "J = carrier R" by simp
1.456 -
1.457 - --{* Calculating an inverse for @{term "a"} *}
1.458 - from one_closed[folded Jcarr]
1.459 - have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
1.460 - by (simp add: J_def r_coset_def set_add_defs)
1.461 - from this
1.462 - obtain r i
1.463 - where rcarr: "r \<in> carrier R"
1.464 - and iI: "i \<in> I"
1.465 - and one: "\<one> = r \<otimes> a \<oplus> i"
1.466 - by fast
1.467 - from one and rcarr and acarr and iI[THEN a_Hcarr]
1.468 - have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
1.469 -
1.470 - --{* Lifting to cosets *}
1.471 - from iI
1.472 - have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
1.473 - by (intro a_rcosI, simp, intro a_subset, simp)
1.474 - from this and rai1
1.475 - have "a \<otimes> r \<in> I +> \<one>" by simp
1.476 - from this have "I +> \<one> = I +> a \<otimes> r"
1.477 - by (rule a_repr_independence, simp) (rule a_subgroup)
1.478 -
1.479 - from rcarr and this[symmetric]
1.480 - show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
1.481 -qed
1.482 qed
1.483
1.484 end