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Theorem subrginv 19274
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 19263 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
21adantr 473 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑅 ∈ Ring)
3 subrginv.1 . . . . . . . 8 𝑆 = (𝑅s 𝐴)
43subrgbas 19267 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5 eqid 2778 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
65subrgss 19259 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
74, 6eqsstr3d 3896 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
87adantr 473 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
93subrgring 19261 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10 subrginv.3 . . . . . . 7 𝑈 = (Unit‘𝑆)
11 subrginv.4 . . . . . . 7 𝐽 = (invr𝑆)
12 eqid 2778 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
1310, 11, 12ringinvcl 19149 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑆))
149, 13sylan 572 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑆))
158, 14sseldd 3859 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑅))
1612, 10unitcl 19132 . . . . . 6 (𝑋𝑈𝑋 ∈ (Base‘𝑆))
1716adantl 474 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑆))
188, 17sseldd 3859 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑅))
19 eqid 2778 . . . . . . 7 (Unit‘𝑅) = (Unit‘𝑅)
203, 19, 10subrguss 19273 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
2120sselda 3858 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Unit‘𝑅))
22 subrginv.2 . . . . . 6 𝐼 = (invr𝑅)
2319, 22, 5ringinvcl 19149 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼𝑋) ∈ (Base‘𝑅))
241, 21, 23syl2an2r 672 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) ∈ (Base‘𝑅))
25 eqid 2778 . . . . 5 (.r𝑅) = (.r𝑅)
265, 25ringass 19037 . . . 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 1352 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))))
28 eqid 2778 . . . . . . 7 (.r𝑆) = (.r𝑆)
29 eqid 2778 . . . . . . 7 (1r𝑆) = (1r𝑆)
3010, 11, 28, 29unitlinv 19150 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑆)𝑋) = (1r𝑆))
319, 30sylan 572 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑆)𝑋) = (1r𝑆))
323, 25ressmulr 16481 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3332adantr 473 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (.r𝑅) = (.r𝑆))
3433oveqd 6993 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)𝑋) = ((𝐽𝑋)(.r𝑆)𝑋))
35 eqid 2778 . . . . . . 7 (1r𝑅) = (1r𝑅)
363, 35subrg1 19268 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
3736adantr 473 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (1r𝑅) = (1r𝑆))
3831, 34, 373eqtr4d 2824 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)𝑋) = (1r𝑅))
3938oveq1d 6991 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((1r𝑅)(.r𝑅)(𝐼𝑋)))
4019, 22, 25, 35unitrinv 19151 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
411, 21, 40syl2an2r 672 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
4241oveq2d 6992 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))) = ((𝐽𝑋)(.r𝑅)(1r𝑅)))
4327, 39, 423eqtr3d 2822 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(1r𝑅)))
445, 25, 35ringlidm 19044 . . 3 ((𝑅 ∈ Ring ∧ (𝐼𝑋) ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = (𝐼𝑋))
451, 24, 44syl2an2r 672 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = (𝐼𝑋))
465, 25, 35ringridm 19045 . . 3 ((𝑅 ∈ Ring ∧ (𝐽𝑋) ∈ (Base‘𝑅)) → ((𝐽𝑋)(.r𝑅)(1r𝑅)) = (𝐽𝑋))
471, 15, 46syl2an2r 672 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)(1r𝑅)) = (𝐽𝑋))
4843, 45, 473eqtr3d 2822 1 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) = (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wss 3829  cfv 6188  (class class class)co 6976  Basecbs 16339  s cress 16340  .rcmulr 16422  1rcur 18974  Ringcrg 19020  Unitcui 19112  invrcinvr 19144  SubRingcsubrg 19254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-tpos 7695  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-2 11503  df-3 11504  df-ndx 16342  df-slot 16343  df-base 16345  df-sets 16346  df-ress 16347  df-plusg 16434  df-mulr 16435  df-0g 16571  df-mgm 17710  df-sgrp 17752  df-mnd 17763  df-grp 17894  df-minusg 17895  df-subg 18060  df-mgp 18963  df-ur 18975  df-ring 19022  df-oppr 19096  df-dvdsr 19114  df-unit 19115  df-invr 19145  df-subrg 19256
This theorem is referenced by:  subrgdv  19275  subrgunit  19276  subrgugrp  19277  issubdrg  19283  gzrngunit  20313
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