Theorem subrginv 14216

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 14205	. . . . 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 14209	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
5		eqid 2229	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
6	5	subrgss 14201	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
7	4, 6	eqsstrrd 3261	. . . . . 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 14203	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		subrginv.3	. . . . . . 7 |- U = (Unit ` S )
11		subrginv.4	. . . . . . 7 |- J = ( invr ` S )
12		eqid 2229	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
13	10, 11, 12	ringinvcl 14104	. . . . . 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 3225	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` R ) )
16		eqidd 2230	. . . . . 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 14025	. . . . . . 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 14087	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` S ) )
23	8, 22	sseldd 3225	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` R ) )
24		eqid 2229	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
25	3, 24, 10	subrguss 14215	. . . . . 6 |- ( A e. (SubRing ` R ) -> U C_ (Unit ` R ) )
26	25	sselda 3224	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. (Unit ` R ) )
27		subrginv.2	. . . . . 6 |- I = ( invr ` R )
28	24, 27, 5	ringinvcl 14104	. . . . 5 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( I ` X ) e. ( Base ` R ) )
29	1, 26, 28	syl2an2r 597	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) e. ( Base ` R ) )
30		eqid 2229	. . . . 5 |- ( .r ` R ) = ( .r ` R )
31	5, 30	ringass 13994	. . . 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 1273	. . 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 2229	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
34		eqid 2229	. . . . . . 7 |- ( 1r ` S ) = ( 1r ` S )
35	10, 11, 33, 34	unitlinv 14105	. . . . . 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 13193	. . . . . . . 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 6024	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( ( J ` X ) ( .r ` S ) X ) )
41		eqid 2229	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
42	3, 41	subrg1 14210	. . . . . 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 2272	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
45	44	oveq1d 6022	. . 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 14106	. . . . 5 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
47	1, 26, 46	syl2an2r 597	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
48	47	oveq2d 6023	. . 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 2270	. 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 14001	. . 3 |- ( ( R e. Ring /\ ( I ` X ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
51	1, 29, 50	syl2an2r 597	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
52	5, 30, 41	ringridm 14002	. . 3 |- ( ( R e. Ring /\ ( J ` X ) e. ( Base ` R ) ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
53	1, 15, 52	syl2an2r 597	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
54	49, 51, 53	3eqtr3d 2270	1 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   C_ wss 3197   ` cfv 5318  (class class class)co 6007   Basecbs 13047   ↾_s cress 13048   .rcmulr 13126   1rcur 13937  SRingcsrg 13941   Ringcrg 13974  Unitcui 14065   invrcinvr 14099  SubRingcsubrg 14196

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-tpos 6397  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-iress 13055  df-plusg 13138  df-mulr 13139  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-minusg 13552  df-subg 13722  df-cmn 13838  df-abl 13839  df-mgp 13899  df-ur 13938  df-srg 13942  df-ring 13976  df-oppr 14046  df-dvdsr 14067  df-unit 14068  df-invr 14100  df-subrg 14198

This theorem is referenced by:  subrgdv  14217  subrgunit  14218  subrgugrp  14219