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Theorem subrgunit 20546
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 20543 . . . 4 (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
54sselda 3976 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝑈)
6 eqid 2725 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
76, 3unitcl 20331 . . . . 5 (𝑋𝑉𝑋 ∈ (Base‘𝑆))
87adantl 480 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋 ∈ (Base‘𝑆))
91subrgbas 20537 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
109adantr 479 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝐴 = (Base‘𝑆))
118, 10eleqtrrd 2828 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → 𝑋𝐴)
121subrgring 20530 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
13 eqid 2725 . . . . . 6 (invr𝑆) = (invr𝑆)
143, 13, 6ringinvcl 20348 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
1512, 14sylan 578 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → ((invr𝑆)‘𝑋) ∈ (Base‘𝑆))
16 subrgunit.4 . . . . 5 𝐼 = (invr𝑅)
171, 16, 3, 13subrginv 20544 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) = ((invr𝑆)‘𝑋))
1815, 17, 103eltr4d 2840 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝐼𝑋) ∈ 𝐴)
195, 11, 183jca 1125 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑉) → (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴))
20 simpr2 1192 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝐴)
219adantr 479 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆))
2220, 21eleqtrd 2827 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
23 simpr3 1193 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ 𝐴)
2423, 21eleqtrd 2827 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝐼𝑋) ∈ (Base‘𝑆))
25 eqid 2725 . . . . . 6 (∥r𝑆) = (∥r𝑆)
26 eqid 2725 . . . . . 6 (.r𝑆) = (.r𝑆)
276, 25, 26dvdsrmul 20320 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
2822, 24, 27syl2anc 582 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)((𝐼𝑋)(.r𝑆)𝑋))
29 subrgrcl 20532 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
3029adantr 479 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑅 ∈ Ring)
31 simpr1 1191 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑈)
32 eqid 2725 . . . . . . 7 (.r𝑅) = (.r𝑅)
33 eqid 2725 . . . . . . 7 (1r𝑅) = (1r𝑅)
342, 16, 32, 33unitlinv 20349 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
3530, 31, 34syl2anc 582 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = (1r𝑅))
361, 32ressmulr 17296 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3736adantr 479 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (.r𝑅) = (.r𝑆))
3837oveqd 7436 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑅)𝑋) = ((𝐼𝑋)(.r𝑆)𝑋))
391, 33subrg1 20538 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
4039adantr 479 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (1r𝑅) = (1r𝑆))
4135, 38, 403eqtr3d 2773 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r𝑆)𝑋) = (1r𝑆))
4228, 41breqtrd 5175 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r𝑆)(1r𝑆))
43 eqid 2725 . . . . . . 7 (oppr𝑆) = (oppr𝑆)
4443, 6opprbas 20297 . . . . . 6 (Base‘𝑆) = (Base‘(oppr𝑆))
45 eqid 2725 . . . . . 6 (∥r‘(oppr𝑆)) = (∥r‘(oppr𝑆))
46 eqid 2725 . . . . . 6 (.r‘(oppr𝑆)) = (.r‘(oppr𝑆))
4744, 45, 46dvdsrmul 20320 . . . . 5 ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
4822, 24, 47syl2anc 582 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))((𝐼𝑋)(.r‘(oppr𝑆))𝑋))
496, 26, 43, 46opprmul 20293 . . . . 5 ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (𝑋(.r𝑆)(𝐼𝑋))
502, 16, 32, 33unitrinv 20350 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5130, 31, 50syl2anc 582 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
5237oveqd 7436 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑅)(𝐼𝑋)) = (𝑋(.r𝑆)(𝐼𝑋)))
5351, 52, 403eqtr3d 2773 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → (𝑋(.r𝑆)(𝐼𝑋)) = (1r𝑆))
5449, 53eqtrid 2777 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → ((𝐼𝑋)(.r‘(oppr𝑆))𝑋) = (1r𝑆))
5548, 54breqtrd 5175 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr𝑆))(1r𝑆))
56 eqid 2725 . . . 4 (1r𝑆) = (1r𝑆)
573, 56, 25, 43, 45isunit 20329 . . 3 (𝑋𝑉 ↔ (𝑋(∥r𝑆)(1r𝑆) ∧ 𝑋(∥r‘(oppr𝑆))(1r𝑆)))
5842, 55, 57sylanbrc 581 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)) → 𝑋𝑉)
5919, 58impbida 799 1 (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17188  s cress 17217  .rcmulr 17242  1rcur 20138  Ringcrg 20190  opprcoppr 20289  rcdsr 20310  Unitcui 20311  invrcinvr 20343  SubRingcsubrg 20523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-nn 12251  df-2 12313  df-3 12314  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17189  df-ress 17218  df-plusg 17254  df-mulr 17255  df-0g 17431  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18906  df-minusg 18907  df-subg 19091  df-cmn 19754  df-abl 19755  df-mgp 20092  df-rng 20110  df-ur 20139  df-ring 20192  df-oppr 20290  df-dvdsr 20313  df-unit 20314  df-invr 20344  df-subrg 20525
This theorem is referenced by:  issubdrg  20685  gzrngunit  21388  zringunit  21414  cphreccllem  25155
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