Theorem subrginv 16880

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 16869	. . . . 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 16873	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
5		eqid 2442	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
6	5	subrgss 16865	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
7	4, 6	eqsstr3d 3390	. . . . . 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 16867	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		subrginv.3	. . . . . . 7 |- U = (Unit ` S )
11		subrginv.4	. . . . . . 7 |- J = ( invr ` S )
12		eqid 2442	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
13	10, 11, 12	rnginvcl 16767	. . . . . 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 3356	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` R ) )
16	12, 10	unitcl 16750	. . . . . 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 3356	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` R ) )
19		eqid 2442	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
20	3, 19, 10	subrguss 16879	. . . . . 6 |- ( A e. (SubRing ` R ) -> U C_ (Unit ` R ) )
21	20	sselda 3355	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. (Unit ` R ) )
22		subrginv.2	. . . . . . 7 |- I = ( invr ` R )
23	19, 22, 5	rnginvcl 16767	. . . . . 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 2442	. . . . 5 |- ( .r ` R ) = ( .r ` R )
27	5, 26	rngass 16660	. . . 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 1220	. . 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 2442	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
30		eqid 2442	. . . . . . 7 |- ( 1r ` S ) = ( 1r ` S )
31	10, 11, 29, 30	unitlinv 16768	. . . . . 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 14290	. . . . . . 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 6107	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( ( J ` X ) ( .r ` S ) X ) )
36		eqid 2442	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
37	3, 36	subrg1 16874	. . . . . 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 2484	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
40	39	oveq1d 6105	. . 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 16769	. . . . . 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 6106	. . 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 2482	. 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 16667	. . 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 16668	. . 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 2482	1 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   C_ wss 3327   ` cfv 5417  (class class class)co 6090   Basecbs 14173   ↾_s cress 14174   .rcmulr 14238   1rcur 16602   Ringcrg 16644  Unitcui 16730   invrcinvr 16762  SubRingcsubrg 16860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-tpos 6744  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-0g 14379  df-mnd 15414  df-grp 15544  df-minusg 15545  df-subg 15677  df-mgp 16591  df-ur 16603  df-rng 16646  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-subrg 16862

This theorem is referenced by:  subrgdv  16881  subrgunit  16882  subrgugrp  16883  issubdrg  16889  gzrngunit  17877