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Theorem subrginv 19018
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 19007 . . . . 5 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
21adantr 472 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑅 ∈ Ring)
3 subrginv.1 . . . . . . . 8 𝑆 = (𝑅s 𝐴)
43subrgbas 19011 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5 eqid 2760 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
65subrgss 19003 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
74, 6eqsstr3d 3781 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
87adantr 472 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
93subrgring 19005 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10 subrginv.3 . . . . . . 7 𝑈 = (Unit‘𝑆)
11 subrginv.4 . . . . . . 7 𝐽 = (invr𝑆)
12 eqid 2760 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
1310, 11, 12ringinvcl 18896 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑆))
149, 13sylan 489 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑆))
158, 14sseldd 3745 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐽𝑋) ∈ (Base‘𝑅))
1612, 10unitcl 18879 . . . . . 6 (𝑋𝑈𝑋 ∈ (Base‘𝑆))
1716adantl 473 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑆))
188, 17sseldd 3745 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑅))
19 eqid 2760 . . . . . . 7 (Unit‘𝑅) = (Unit‘𝑅)
203, 19, 10subrguss 19017 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
2120sselda 3744 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → 𝑋 ∈ (Unit‘𝑅))
22 subrginv.2 . . . . . . 7 𝐼 = (invr𝑅)
2319, 22, 5ringinvcl 18896 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼𝑋) ∈ (Base‘𝑅))
241, 23sylan 489 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼𝑋) ∈ (Base‘𝑅))
2521, 24syldan 488 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) ∈ (Base‘𝑅))
26 eqid 2760 . . . . 5 (.r𝑅) = (.r𝑅)
275, 26ringass 18784 . . . 4 ((𝑅 ∈ Ring ∧ ((𝐽𝑋) ∈ (Base‘𝑅) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝐼𝑋) ∈ (Base‘𝑅))) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))))
282, 15, 18, 25, 27syl13anc 1479 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))))
29 eqid 2760 . . . . . . 7 (.r𝑆) = (.r𝑆)
30 eqid 2760 . . . . . . 7 (1r𝑆) = (1r𝑆)
3110, 11, 29, 30unitlinv 18897 . . . . . 6 ((𝑆 ∈ Ring ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑆)𝑋) = (1r𝑆))
329, 31sylan 489 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑆)𝑋) = (1r𝑆))
333, 26ressmulr 16228 . . . . . . 7 (𝐴 ∈ (SubRing‘𝑅) → (.r𝑅) = (.r𝑆))
3433adantr 472 . . . . . 6 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (.r𝑅) = (.r𝑆))
3534oveqd 6831 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)𝑋) = ((𝐽𝑋)(.r𝑆)𝑋))
36 eqid 2760 . . . . . . 7 (1r𝑅) = (1r𝑅)
373, 36subrg1 19012 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → (1r𝑅) = (1r𝑆))
3837adantr 472 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (1r𝑅) = (1r𝑆))
3932, 35, 383eqtr4d 2804 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)𝑋) = (1r𝑅))
4039oveq1d 6829 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (((𝐽𝑋)(.r𝑅)𝑋)(.r𝑅)(𝐼𝑋)) = ((1r𝑅)(.r𝑅)(𝐼𝑋)))
4119, 22, 26, 36unitrinv 18898 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
421, 41sylan 489 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
4321, 42syldan 488 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝑋(.r𝑅)(𝐼𝑋)) = (1r𝑅))
4443oveq2d 6830 . . 3 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)(𝑋(.r𝑅)(𝐼𝑋))) = ((𝐽𝑋)(.r𝑅)(1r𝑅)))
4528, 40, 443eqtr3d 2802 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = ((𝐽𝑋)(.r𝑅)(1r𝑅)))
465, 26, 36ringlidm 18791 . . 3 ((𝑅 ∈ Ring ∧ (𝐼𝑋) ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = (𝐼𝑋))
472, 25, 46syl2anc 696 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((1r𝑅)(.r𝑅)(𝐼𝑋)) = (𝐼𝑋))
485, 26, 36ringridm 18792 . . 3 ((𝑅 ∈ Ring ∧ (𝐽𝑋) ∈ (Base‘𝑅)) → ((𝐽𝑋)(.r𝑅)(1r𝑅)) = (𝐽𝑋))
492, 15, 48syl2anc 696 . 2 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → ((𝐽𝑋)(.r𝑅)(1r𝑅)) = (𝐽𝑋))
5045, 47, 493eqtr3d 2802 1 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) = (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wss 3715  cfv 6049  (class class class)co 6814  Basecbs 16079  s cress 16080  .rcmulr 16164  1rcur 18721  Ringcrg 18767  Unitcui 18859  invrcinvr 18891  SubRingcsubrg 18998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-tpos 7522  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-mulr 16177  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-subg 17812  df-mgp 18710  df-ur 18722  df-ring 18769  df-oppr 18843  df-dvdsr 18861  df-unit 18862  df-invr 18892  df-subrg 19000
This theorem is referenced by:  subrgdv  19019  subrgunit  19020  subrgugrp  19021  issubdrg  19027  gzrngunit  20034
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