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Theorem subrginv 19547
Description: A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrginv.1 𝑆 = (𝑅s 𝐴)
subrginv.2 𝐼 = (invr𝑅)
subrginv.3 𝑈 = (Unit‘𝑆)
subrginv.4 𝐽 = (invr𝑆)
Assertion
Ref Expression
subrginv ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) = (𝐽𝑋))

Proof of Theorem subrginv
StepHypRef Expression
1 subrgrcl 19536 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
21adantr 484 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑅 ∈ Ring)
3 subrginv.1 . . . . . . . 8 𝑆 = (𝑅s 𝐴)
43subrgbas 19540 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5 eqid 2801 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
65subrgss 19532 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
74, 6eqsstrrd 3957 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
87adantr 484 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
93subrgring 19534 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10 subrginv.3 . . . . . . 7 𝑈 = (Unit‘𝑆)
11 subrginv.4 . . . . . . 7 𝐽 = (invr𝑆)
12 eqid 2801 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
1310, 11, 12ringinvcl 19425 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑆))
149, 13sylan 583 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑆))
158, 14sseldd 3919 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑅))
1612, 10unitcl 19408 . . . . . 6 (𝑋𝑈𝑋 ∈ (Base‘𝑆))
1716adantl 485 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑆))
188, 17sseldd 3919 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑅))
19 eqid 2801 . . . . . . 7 (Unit‘𝑅) = (Unit‘𝑅)
203, 19, 10subrguss 19546 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
2120sselda 3918 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Unit‘𝑅))
22 subrginv.2 . . . . . 6 𝐼 = (invr𝑅)
2319, 22, 5ringinvcl 19425 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼𝑋) ∈ (Base‘𝑅))
241, 21, 23syl2an2r 684 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) ∈ (Base‘𝑅))
25 eqid 2801 . . . . 5 (.r𝑅) = (.r𝑅)
265, 25ringass 19313 . . . 4 ((𝑅 ∈ Ring ∧ ((𝐽𝑋) ∈ (Base‘𝑅) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝐼𝑋) ∈ (Base‘𝑅))) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))))
272, 15, 18, 24, 26syl13anc 1369 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))))
28 eqid 2801 . . . . . . 7 (.r𝑆) = (.r𝑆)
29 eqid 2801 . . . . . . 7 (1r𝑆) = (1r𝑆)
3010, 11, 28, 29unitlinv 19426 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑆)𝑋) = (1r𝑆))
319, 30sylan 583 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑆)𝑋) = (1r𝑆))
323, 25ressmulr 16620 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3332adantr 484 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (.r𝑅) = (.r𝑆))
3433oveqd 7156 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)𝑋) = ((𝐽𝑋)(.r𝑆)𝑋))
35 eqid 2801 . . . . . . 7 (1r𝑅) = (1r𝑅)
363, 35subrg1 19541 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
3736adantr 484 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (1r𝑅) = (1r𝑆))
3831, 34, 373eqtr4d 2846 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)𝑋) = (1r𝑅))
3938oveq1d 7154 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((1r𝑅)(.r𝑅)(𝐼𝑋)))
4019, 22, 25, 35unitrinv 19427 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
411, 21, 40syl2an2r 684 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
4241oveq2d 7155 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))) = ((𝐽𝑋)(.r𝑅)(1r𝑅)))
4327, 39, 423eqtr3d 2844 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(1r𝑅)))
445, 25, 35ringlidm 19320 . . 3 ((𝑅 ∈ Ring ∧ (𝐼𝑋) ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = (𝐼𝑋))
451, 24, 44syl2an2r 684 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = (𝐼𝑋))
465, 25, 35ringridm 19321 . . 3 ((𝑅 ∈ Ring ∧ (𝐽𝑋) ∈ (Base‘𝑅)) → ((𝐽𝑋)(.r𝑅)(1r𝑅)) = (𝐽𝑋))
471, 15, 46syl2an2r 684 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)(1r𝑅)) = (𝐽𝑋))
4843, 45, 473eqtr3d 2844 1 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) = (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wss 3884  cfv 6328  (class class class)co 7139  Basecbs 16478  s cress 16479  .rcmulr 16561  1rcur 19247  Ringcrg 19293  Unitcui 19388  invrcinvr 19420  SubRingcsubrg 19527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-minusg 18102  df-subg 18271  df-mgp 19236  df-ur 19248  df-ring 19295  df-oppr 19372  df-dvdsr 19390  df-unit 19391  df-invr 19421  df-subrg 19529
This theorem is referenced by:  subrgdv  19548  subrgunit  19549  subrgugrp  19550  issubdrg  19556  gzrngunit  20160
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