Theorem subrginv 18036

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 18025	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. Ring )
2	1	adantr 467	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> R e. Ring )
3		subrginv.1	. . . . . . . 8 |- S = ( R ↾s A )
4	3	subrgbas 18029	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
5		eqid 2453	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
6	5	subrgss 18021	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
7	4, 6	eqsstr3d 3469	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
8	7	adantr 467	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( Base ` S ) C_ ( Base ` R ) )
9	3	subrgring 18023	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		subrginv.3	. . . . . . 7 |- U = (Unit ` S )
11		subrginv.4	. . . . . . 7 |- J = ( invr ` S )
12		eqid 2453	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
13	10, 11, 12	ringinvcl 17916	. . . . . 6 |- ( ( S e. Ring /\ X e. U ) -> ( J ` X ) e. ( Base ` S ) )
14	9, 13	sylan 474	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` S ) )
15	8, 14	sseldd 3435	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( J ` X ) e. ( Base ` R ) )
16	12, 10	unitcl 17899	. . . . . 6 |- ( X e. U -> X e. ( Base ` S ) )
17	16	adantl 468	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` S ) )
18	8, 17	sseldd 3435	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. ( Base ` R ) )
19		eqid 2453	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
20	3, 19, 10	subrguss 18035	. . . . . 6 |- ( A e. (SubRing ` R ) -> U C_ (Unit ` R ) )
21	20	sselda 3434	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> X e. (Unit ` R ) )
22		subrginv.2	. . . . . . 7 |- I = ( invr ` R )
23	19, 22, 5	ringinvcl 17916	. . . . . 6 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( I ` X ) e. ( Base ` R ) )
24	1, 23	sylan 474	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. (Unit ` R ) ) -> ( I ` X ) e. ( Base ` R ) )
25	21, 24	syldan 473	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) e. ( Base ` R ) )
26		eqid 2453	. . . . 5 |- ( .r ` R ) = ( .r ` R )
27	5, 26	ringass 17809	. . . 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 1271	. . 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 2453	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
30		eqid 2453	. . . . . . 7 |- ( 1r ` S ) = ( 1r ` S )
31	10, 11, 29, 30	unitlinv 17917	. . . . . 6 |- ( ( S e. Ring /\ X e. U ) -> ( ( J ` X ) ( .r ` S ) X ) = ( 1r ` S ) )
32	9, 31	sylan 474	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` S ) X ) = ( 1r ` S ) )
33	3, 26	ressmulr 15262	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
34	33	adantr 467	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( .r ` R ) = ( .r ` S ) )
35	34	oveqd 6312	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( ( J ` X ) ( .r ` S ) X ) )
36		eqid 2453	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
37	3, 36	subrg1 18030	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` S ) )
38	37	adantr 467	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( 1r ` R ) = ( 1r ` S ) )
39	32, 35, 38	3eqtr4d 2497	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
40	39	oveq1d 6310	. . 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 17918	. . . . . 6 |- ( ( R e. Ring /\ X e. (Unit ` R ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
42	1, 41	sylan 474	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ X e. (Unit ` R ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
43	21, 42	syldan 473	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
44	43	oveq2d 6311	. . 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 2495	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) )
46	5, 26, 36	ringlidm 17816	. . 3 |- ( ( R e. Ring /\ ( I ` X ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
47	2, 25, 46	syl2anc 667	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( 1r ` R ) ( .r ` R ) ( I ` X ) ) = ( I ` X ) )
48	5, 26, 36	ringridm 17817	. . 3 |- ( ( R e. Ring /\ ( J ` X ) e. ( Base ` R ) ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
49	2, 15, 48	syl2anc 667	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( ( J ` X ) ( .r ` R ) ( 1r ` R ) ) = ( J ` X ) )
50	45, 47, 49	3eqtr3d 2495	1 |- ( ( A e. (SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   C_ wss 3406   ` cfv 5585  (class class class)co 6295   Basecbs 15133   ↾_s cress 15134   .rcmulr 15203   1rcur 17747   Ringcrg 17792  Unitcui 17879   invrcinvr 17911  SubRingcsubrg 18016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-0g 15352  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-grp 16685  df-minusg 16686  df-subg 16826  df-mgp 17736  df-ur 17748  df-ring 17794  df-oppr 17863  df-dvdsr 17881  df-unit 17882  df-invr 17912  df-subrg 18018

This theorem is referenced by:  subrgdv  18037  subrgunit  18038  subrgugrp  18039  issubdrg  18045  gzrngunit  19045