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Theorem subrginv 20038
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 20027 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
21adantr 481 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑅 ∈ Ring)
3 subrginv.1 . . . . . . . 8 𝑆 = (𝑅s 𝐴)
43subrgbas 20031 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5 eqid 2740 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
65subrgss 20023 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
74, 6eqsstrrd 3965 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
87adantr 481 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
93subrgring 20025 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10 subrginv.3 . . . . . . 7 𝑈 = (Unit‘𝑆)
11 subrginv.4 . . . . . . 7 𝐽 = (invr𝑆)
12 eqid 2740 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
1310, 11, 12ringinvcl 19916 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑆))
149, 13sylan 580 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑆))
158, 14sseldd 3927 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑅))
1612, 10unitcl 19899 . . . . . 6 (𝑋𝑈𝑋 ∈ (Base‘𝑆))
1716adantl 482 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑆))
188, 17sseldd 3927 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑅))
19 eqid 2740 . . . . . . 7 (Unit‘𝑅) = (Unit‘𝑅)
203, 19, 10subrguss 20037 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
2120sselda 3926 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Unit‘𝑅))
22 subrginv.2 . . . . . 6 𝐼 = (invr𝑅)
2319, 22, 5ringinvcl 19916 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼𝑋) ∈ (Base‘𝑅))
241, 21, 23syl2an2r 682 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) ∈ (Base‘𝑅))
25 eqid 2740 . . . . 5 (.r𝑅) = (.r𝑅)
265, 25ringass 19801 . . . 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 1371 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))))
28 eqid 2740 . . . . . . 7 (.r𝑆) = (.r𝑆)
29 eqid 2740 . . . . . . 7 (1r𝑆) = (1r𝑆)
3010, 11, 28, 29unitlinv 19917 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑆)𝑋) = (1r𝑆))
319, 30sylan 580 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑆)𝑋) = (1r𝑆))
323, 25ressmulr 17015 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3332adantr 481 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (.r𝑅) = (.r𝑆))
3433oveqd 7288 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)𝑋) = ((𝐽𝑋)(.r𝑆)𝑋))
35 eqid 2740 . . . . . . 7 (1r𝑅) = (1r𝑅)
363, 35subrg1 20032 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
3736adantr 481 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (1r𝑅) = (1r𝑆))
3831, 34, 373eqtr4d 2790 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)𝑋) = (1r𝑅))
3938oveq1d 7286 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((1r𝑅)(.r𝑅)(𝐼𝑋)))
4019, 22, 25, 35unitrinv 19918 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
411, 21, 40syl2an2r 682 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
4241oveq2d 7287 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))) = ((𝐽𝑋)(.r𝑅)(1r𝑅)))
4327, 39, 423eqtr3d 2788 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(1r𝑅)))
445, 25, 35ringlidm 19808 . . 3 ((𝑅 ∈ Ring ∧ (𝐼𝑋) ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = (𝐼𝑋))
451, 24, 44syl2an2r 682 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = (𝐼𝑋))
465, 25, 35ringridm 19809 . . 3 ((𝑅 ∈ Ring ∧ (𝐽𝑋) ∈ (Base‘𝑅)) → ((𝐽𝑋)(.r𝑅)(1r𝑅)) = (𝐽𝑋))
471, 15, 46syl2an2r 682 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)(1r𝑅)) = (𝐽𝑋))
4843, 45, 473eqtr3d 2788 1 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) = (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1542  wcel 2110  wss 3892  cfv 6432  (class class class)co 7271  Basecbs 16910  s cress 16939  .rcmulr 16961  1rcur 19735  Ringcrg 19781  Unitcui 19879  invrcinvr 19911  SubRingcsubrg 20018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-2nd 7825  df-tpos 8033  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-sets 16863  df-slot 16881  df-ndx 16893  df-base 16911  df-ress 16940  df-plusg 16973  df-mulr 16974  df-0g 17150  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-grp 18578  df-minusg 18579  df-subg 18750  df-mgp 19719  df-ur 19736  df-ring 19783  df-oppr 19860  df-dvdsr 19881  df-unit 19882  df-invr 19912  df-subrg 20020
This theorem is referenced by:  subrgdv  20039  subrgunit  20040  subrgugrp  20041  issubdrg  20047  gzrngunit  20662
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