Theorem subrginv 17223

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 = ( R ↾s A )
subrginv.2	|- I = ( invr ` R )
subrginv.3	|- U = (Unit ` S )
subrginv.4	|- J = ( invr ` S )

Assertion

Ref	Expression
subrginv	|- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) )

Proof of Theorem subrginv

Step	Hyp	Ref	Expression
1		subrgrcl 17212	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. Ring )
2	1	adantr 465	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> R e. Ring )
3		subrginv.1	. . . . . . . 8 |- S = ( R ↾s A )
4	3	subrgbas 17216	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
5		eqid 2462	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
6	5	subrgss 17208	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
7	4, 6	eqsstr3d 3534	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
8	7	adantr 465	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( Base ` S ) C_ ( Base ` R ) )
9	3	subrgrng 17210	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		subrginv.3	. . . . . . 7 |- U = (Unit ` S )
11		subrginv.4	. . . . . . 7 |- J = ( invr ` S )
12		eqid 2462	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
13	10, 11, 12	rnginvcl 17104	. . . . . 6 |- ( ( S e. Ring /\ X e. U ) -> ( J ` X ) e. ( Base ` S ) )
14	9, 13	sylan 471	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` S ) )
15	8, 14	sseldd 3500	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` R ) )
16	12, 10	unitcl 17087	. . . . . 6 |- ( X e. U -> X e. ( Base ` S ) )
17	16	adantl 466	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` S ) )
18	8, 17	sseldd 3500	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` R ) )
19		eqid 2462	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
20	3, 19, 10	subrguss 17222	. . . . . 6 |- ( A e. (SubRing ` R ) -> U C_ (Unit ` R ) )
21	20	sselda 3499	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. (Unit ` R ) )
22		subrginv.2	. . . . . . 7 |- I = ( invr ` R )
23	19, 22, 5	rnginvcl 17104	. . . . . 6 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( I ` X ) e. ( Base ` R ) )
24	1, 23	sylan 471	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. (Unit ` R ) ) -> ( I ` X ) e. ( Base ` R ) )
25	21, 24	syldan 470	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) e. ( Base ` R ) )
26		eqid 2462	. . . . 5 |- ( .r ` R ) = ( .r ` R )
27	5, 26	rngass 16997	. . . 4 |- ( ( R e. Ring /\ ( ( J ` X ) e. ( Base ` R ) /\ X e. ( Base ` R ) /\ ( I ` X ) e. ( Base ` R ) ) ) -> ( ( ( J ` X ) ( .r ` R ) X ) ( .r ` R ) ( I ` X ) ) = ( ( J ` X ) ( .r ` R ) ( X ( .r ` R ) ( I ` X ) ) ) )
28	2, 15, 18, 25, 27	syl13anc 1225	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( ( J ` X ) ( .r ` R ) X ) ( .r ` R ) ( I ` X ) ) = ( ( J ` X ) ( .r ` R ) ( X ( .r ` R ) ( I ` X ) ) ) )
29		eqid 2462	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
30		eqid 2462	. . . . . . 7 |- ( 1r ` S ) = ( 1r ` S )
31	10, 11, 29, 30	unitlinv 17105	. . . . . 6 |- ( ( S e. Ring /\ X e. U ) -> ( ( J ` X ) ( .r ` S ) X ) = ( 1r ` S ) )
32	9, 31	sylan 471	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` S ) X ) = ( 1r ` S ) )
33	3, 26	ressmulr 14599	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
34	33	adantr 465	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( .r ` R ) = ( .r ` S ) )
35	34	oveqd 6294	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( ( J ` X ) ( .r ` S ) X ) )
36		eqid 2462	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
37	3, 36	subrg1 17217	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` S ) )
38	37	adantr 465	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( 1r ` R ) = ( 1r ` S ) )
39	32, 35, 38	3eqtr4d 2513	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
40	39	oveq1d 6292	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( ( J ` X ) ( .r ` R ) X ) ( .r ` R ) ( I ` X ) ) = ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) )
41	19, 22, 26, 36	unitrinv 17106	. . . . . 6 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
42	1, 41	sylan 471	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. (Unit ` R ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
43	21, 42	syldan 470	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
44	43	oveq2d 6293	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) ( X ( .r ` R ) ( I ` X ) ) ) = ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) )
45	28, 40, 44	3eqtr3d 2511	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) )
46	5, 26, 36	rnglidm 17004	. . 3 |- ( ( R e. Ring /\ ( I ` X ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
47	2, 25, 46	syl2anc 661	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
48	5, 26, 36	rngridm 17005	. . 3 |- ( ( R e. Ring /\ ( J ` X ) e. ( Base ` R ) ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
49	2, 15, 48	syl2anc 661	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
50	45, 47, 49	3eqtr3d 2511	1 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   C_ wss 3471   ` cfv 5581  (class class class)co 6277   Basecbs 14481   ↾_s cress 14482   .rcmulr 14547   1rcur 16938   Ringcrg 16981  Unitcui 17067   invrcinvr 17099  SubRingcsubrg 17203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-0g 14688  df-mnd 15723  df-grp 15853  df-minusg 15854  df-subg 15988  df-mgp 16927  df-ur 16939  df-rng 16983  df-oppr 17051  df-dvdsr 17069  df-unit 17070  df-invr 17100  df-subrg 17205

This theorem is referenced by:  subrgdv  17224  subrgunit  17225  subrgugrp  17226  issubdrg  17232  gzrngunit  18246