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Theorem subrgunit 19070
Description: An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrgugrp.1 𝑆 = (𝑅s 𝐴)
subrgugrp.2 𝑈 = (Unit‘𝑅)
subrgugrp.3 𝑉 = (Unit‘𝑆)
subrgunit.4 𝐼 = (invr𝑅)
Assertion
Ref Expression
subrgunit (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))

Proof of Theorem subrgunit
StepHypRef Expression
1 subrgugrp.1 . . . . 5 𝑆 = (𝑅s 𝐴)
2 subrgugrp.2 . . . . 5 𝑈 = (Unit‘𝑅)
3 subrgugrp.3 . . . . 5 𝑉 = (Unit‘𝑆)
41, 2, 3subrguss 19067 . . . 4 (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
54sselda 3763 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝑈)
6 eqid 2765 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
76, 3unitcl 18929 . . . . 5 (𝑋𝑉𝑋 ∈ (Base‘𝑆))
87adantl 473 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋 ∈ (Base‘𝑆))
91subrgbas 19061 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
109adantr 472 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝐴 = (Base‘𝑆))
118, 10eleqtrrd 2847 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝐴)
121subrgring 19055 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
13 eqid 2765 . . . . . 6 (invr𝑆) = (invr𝑆)
143, 13, 6ringinvcl 18946 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
1512, 14sylan 575 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
16 subrgunit.4 . . . . 5 𝐼 = (invr𝑅)
171, 16, 3, 13subrginv 19068 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) = ((invr𝑆)‘𝑋))
1815, 17, 103eltr4d 2859 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) ∈ 𝐴)
195, 11, 183jca 1158 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴))
20 simpr2 1250 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝐴)
219adantr 472 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆))
2220, 21eleqtrd 2846 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
23 simpr3 1252 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ 𝐴)
2423, 21eleqtrd 2846 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ (Base‘𝑆))
25 eqid 2765 . . . . . 6 (∥r𝑆) = (∥r𝑆)
26 eqid 2765 . . . . . 6 (.r𝑆) = (.r𝑆)
276, 25, 26dvdsrmul 18918 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
2822, 24, 27syl2anc 579 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
29 subrgrcl 19057 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3029adantr 472 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑅 ∈ Ring)
31 simpr1 1248 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑈)
32 eqid 2765 . . . . . . 7 (.r𝑅) = (.r𝑅)
33 eqid 2765 . . . . . . 7 (1r𝑅) = (1r𝑅)
342, 16, 32, 33unitlinv 18947 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
3530, 31, 34syl2anc 579 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
361, 32ressmulr 16281 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3736adantr 472 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (.r𝑅) = (.r𝑆))
3837oveqd 6861 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = ((𝐼𝑋)(.r𝑆)𝑋))
391, 33subrg1 19062 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
4039adantr 472 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (1r𝑅) = (1r𝑆))
4135, 38, 403eqtr3d 2807 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑆)𝑋) = (1r𝑆))
4228, 41breqtrd 4837 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)(1r𝑆))
43 eqid 2765 . . . . . . 7 (oppr𝑆) = (oppr𝑆)
4443, 6opprbas 18899 . . . . . 6 (Base‘𝑆) = (Base‘(oppr𝑆))
45 eqid 2765 . . . . . 6 (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆))
46 eqid 2765 . . . . . 6 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
4744, 45, 46dvdsrmul 18918 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
4822, 24, 47syl2anc 579 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
496, 26, 43, 46opprmul 18896 . . . . 5 ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (𝑋(.r𝑆)(𝐼𝑋))
502, 16, 32, 33unitrinv 18948 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5130, 31, 50syl2anc 579 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5237oveqd 6861 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (𝑋(.r𝑆)(𝐼𝑋)))
5351, 52, 403eqtr3d 2807 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑆)(𝐼𝑋)) = (1r𝑆))
5449, 53syl5eq 2811 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (1r𝑆))
5548, 54breqtrd 4837 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))(1r𝑆))
56 eqid 2765 . . . 4 (1r𝑆) = (1r𝑆)
573, 56, 25, 43, 45isunit 18927 . . 3 (𝑋𝑉 ↔ (𝑋(∥r𝑆)(1r𝑆) ∧ 𝑋(∥r‘(oppr𝑆))(1r𝑆)))
5842, 55, 57sylanbrc 578 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑉)
5919, 58impbida 835 1 (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155   class class class wbr 4811  cfv 6070  (class class class)co 6844  Basecbs 16133  s cress 16134  .rcmulr 16218  1rcur 18771  Ringcrg 18817  opprcoppr 18892  rcdsr 18908  Unitcui 18909  invrcinvr 18941  SubRingcsubrg 19048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-tpos 7557  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-3 11338  df-ndx 16136  df-slot 16137  df-base 16139  df-sets 16140  df-ress 16141  df-plusg 16230  df-mulr 16231  df-0g 16371  df-mgm 17511  df-sgrp 17553  df-mnd 17564  df-grp 17695  df-minusg 17696  df-subg 17858  df-mgp 18760  df-ur 18772  df-ring 18819  df-oppr 18893  df-dvdsr 18911  df-unit 18912  df-invr 18942  df-subrg 19050
This theorem is referenced by:  issubdrg  19077  gzrngunit  20088  zringunit  20112  cphreccllem  23259
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