Theorem subrginv 13917

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 13906	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. Ring )
2	1	adantr 276	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> R e. Ring )
3		subrginv.1	. . . . . . . 8 |- S = ( R ↾s A )
4	3	subrgbas 13910	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
5		eqid 2204	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
6	5	subrgss 13902	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
7	4, 6	eqsstrrd 3229	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
8	7	adantr 276	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( Base ` S ) C_ ( Base ` R ) )
9	3	subrgring 13904	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		subrginv.3	. . . . . . 7 |- U = (Unit ` S )
11		subrginv.4	. . . . . . 7 |- J = ( invr ` S )
12		eqid 2204	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
13	10, 11, 12	ringinvcl 13805	. . . . . 6 |- ( ( S e. Ring /\ X e. U ) -> ( J ` X ) e. ( Base ` S ) )
14	9, 13	sylan 283	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` S ) )
15	8, 14	sseldd 3193	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` R ) )
16		eqidd 2205	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( Base ` S ) = ( Base ` S ) )
17	10	a1i 9	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> U = (Unit ` S ) )
18	9	adantr 276	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> S e. Ring )
19		ringsrg 13727	. . . . . . 7 |- ( S e. Ring -> S e. SRing )
20	18, 19	syl 14	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> S e. SRing )
21		simpr 110	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. U )
22	16, 17, 20, 21	unitcld 13788	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` S ) )
23	8, 22	sseldd 3193	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` R ) )
24		eqid 2204	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
25	3, 24, 10	subrguss 13916	. . . . . 6 |- ( A e. (SubRing ` R ) -> U C_ (Unit ` R ) )
26	25	sselda 3192	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. (Unit ` R ) )
27		subrginv.2	. . . . . 6 |- I = ( invr ` R )
28	24, 27, 5	ringinvcl 13805	. . . . 5 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( I ` X ) e. ( Base ` R ) )
29	1, 26, 28	syl2an2r 595	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) e. ( Base ` R ) )
30		eqid 2204	. . . . 5 |- ( .r ` R ) = ( .r ` R )
31	5, 30	ringass 13696	. . . 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 ) ) ) )
32	2, 15, 23, 29, 31	syl13anc 1251	. . 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 ) ) ) )
33		eqid 2204	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
34		eqid 2204	. . . . . . 7 |- ( 1r ` S ) = ( 1r ` S )
35	10, 11, 33, 34	unitlinv 13806	. . . . . 6 |- ( ( S e. Ring /\ X e. U ) -> ( ( J ` X ) ( .r ` S ) X ) = ( 1r ` S ) )
36	9, 35	sylan 283	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` S ) X ) = ( 1r ` S ) )
37	3, 30	ressmulrg 12895	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
38	1, 37	mpdan 421	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
39	38	adantr 276	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( .r ` R ) = ( .r ` S ) )
40	39	oveqd 5951	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( ( J ` X ) ( .r ` S ) X ) )
41		eqid 2204	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
42	3, 41	subrg1 13911	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` S ) )
43	42	adantr 276	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( 1r ` R ) = ( 1r ` S ) )
44	36, 40, 43	3eqtr4d 2247	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
45	44	oveq1d 5949	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( ( J ` X ) ( .r ` R ) X ) ( .r ` R ) ( I ` X ) ) = ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) )
46	24, 27, 30, 41	unitrinv 13807	. . . . 5 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
47	1, 26, 46	syl2an2r 595	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
48	47	oveq2d 5950	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) ( X ( .r ` R ) ( I ` X ) ) ) = ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) )
49	32, 45, 48	3eqtr3d 2245	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) )
50	5, 30, 41	ringlidm 13703	. . 3 |- ( ( R e. Ring /\ ( I ` X ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
51	1, 29, 50	syl2an2r 595	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
52	5, 30, 41	ringridm 13704	. . 3 |- ( ( R e. Ring /\ ( J ` X ) e. ( Base ` R ) ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
53	1, 15, 52	syl2an2r 595	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
54	49, 51, 53	3eqtr3d 2245	1 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   C_ wss 3165   ` cfv 5268  (class class class)co 5934   Basecbs 12751   ↾_s cress 12752   .rcmulr 12829   1rcur 13639  SRingcsrg 13643   Ringcrg 13676  Unitcui 13767   invrcinvr 13800  SubRingcsubrg 13897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-addcom 8007  ax-addass 8009  ax-i2m1 8012  ax-0lt1 8013  ax-0id 8015  ax-rnegex 8016  ax-pre-ltirr 8019  ax-pre-lttrn 8021  ax-pre-ltadd 8023

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-tpos 6321  df-pnf 8091  df-mnf 8092  df-ltxr 8094  df-inn 9019  df-2 9077  df-3 9078  df-ndx 12754  df-slot 12755  df-base 12757  df-sets 12758  df-iress 12759  df-plusg 12841  df-mulr 12842  df-0g 13008  df-mgm 13106  df-sgrp 13152  df-mnd 13167  df-grp 13253  df-minusg 13254  df-subg 13424  df-cmn 13540  df-abl 13541  df-mgp 13601  df-ur 13640  df-srg 13644  df-ring 13678  df-oppr 13748  df-dvdsr 13769  df-unit 13770  df-invr 13801  df-subrg 13899

This theorem is referenced by:  subrgdv  13918  subrgunit  13919  subrgugrp  13920