Theorem subrginv 14382

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 14371	. . . . 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 14375	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
5		eqid 2232	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
6	5	subrgss 14367	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
7	4, 6	eqsstrrd 3275	. . . . . 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 14369	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		subrginv.3	. . . . . . 7 |- U = (Unit ` S )
11		subrginv.4	. . . . . . 7 |- J = ( invr ` S )
12		eqid 2232	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
13	10, 11, 12	ringinvcl 14270	. . . . . 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 3239	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` R ) )
16		eqidd 2233	. . . . . 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 14191	. . . . . . 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 14253	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` S ) )
23	8, 22	sseldd 3239	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` R ) )
24		eqid 2232	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
25	3, 24, 10	subrguss 14381	. . . . . 6 |- ( A e. (SubRing ` R ) -> U C_ (Unit ` R ) )
26	25	sselda 3238	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. (Unit ` R ) )
27		subrginv.2	. . . . . 6 |- I = ( invr ` R )
28	24, 27, 5	ringinvcl 14270	. . . . 5 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( I ` X ) e. ( Base ` R ) )
29	1, 26, 28	syl2an2r 599	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) e. ( Base ` R ) )
30		eqid 2232	. . . . 5 |- ( .r ` R ) = ( .r ` R )
31	5, 30	ringass 14160	. . . 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 1276	. . 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 2232	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
34		eqid 2232	. . . . . . 7 |- ( 1r ` S ) = ( 1r ` S )
35	10, 11, 33, 34	unitlinv 14271	. . . . . 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 13358	. . . . . . . 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 6067	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( ( J ` X ) ( .r ` S ) X ) )
41		eqid 2232	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
42	3, 41	subrg1 14376	. . . . . 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 2275	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
45	44	oveq1d 6065	. . 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 14272	. . . . 5 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
47	1, 26, 46	syl2an2r 599	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
48	47	oveq2d 6066	. . 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 2273	. 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 14167	. . 3 |- ( ( R e. Ring /\ ( I ` X ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
51	1, 29, 50	syl2an2r 599	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
52	5, 30, 41	ringridm 14168	. . 3 |- ( ( R e. Ring /\ ( J ` X ) e. ( Base ` R ) ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
53	1, 15, 52	syl2an2r 599	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
54	49, 51, 53	3eqtr3d 2273	1 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   C_ wss 3211   ` cfv 5352  (class class class)co 6050   Basecbs 13212   ↾_s cress 13213   .rcmulr 13291   1rcur 14103  SRingcsrg 14107   Ringcrg 14140  Unitcui 14231   invrcinvr 14265  SubRingcsubrg 14362

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-tpos 6476  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-subg 13887  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-oppr 14212  df-dvdsr 14233  df-unit 14234  df-invr 14266  df-subrg 14364

This theorem is referenced by:  subrgdv  14383  subrgunit  14384  subrgugrp  14385