Theorem subrg1 13312

Description: A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Hypotheses

Ref	Expression
subrg1.1	|- S = ( R ↾s A )
subrg1.2	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
subrg1	|- ( A e. (SubRing ` R ) -> .1. = ( 1r ` S ) )

Proof of Theorem subrg1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrg1.2	. 2 |- .1. = ( 1r ` R )
2		eqid 2177	. . . . 5 |- ( 1r ` R ) = ( 1r ` R )
3	2	subrg1cl 13310	. . . 4 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
4		subrg1.1	. . . . 5 |- S = ( R ↾s A )
5	4	subrgbas 13311	. . . 4 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
6	3, 5	eleqtrd 2256	. . 3 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. ( Base ` S ) )
7		eqid 2177	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
8	7	subrgss 13303	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
9	5, 8	eqsstrrd 3192	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
10	9	sselda 3155	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` S ) ) -> x e. ( Base ` R ) )
11		subrgrcl 13307	. . . . . . 7 |- ( A e. (SubRing ` R ) -> R e. Ring )
12		eqid 2177	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
13	7, 12, 2	ringidmlem 13158	. . . . . . 7 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( ( 1r ` R ) ( .r ` R ) x ) = x /\ ( x ( .r ` R ) ( 1r ` R ) ) = x ) )
14	11, 13	sylan 283	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` R ) ) -> ( ( ( 1r ` R ) ( .r ` R ) x ) = x /\ ( x ( .r ` R ) ( 1r ` R ) ) = x ) )
15	4, 12	ressmulrg 12597	. . . . . . . . . . 11 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
16	11, 15	mpdan 421	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
17	16	oveqd 5891	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( ( 1r ` R ) ( .r ` R ) x ) = ( ( 1r ` R ) ( .r ` S ) x ) )
18	17	eqeq1d 2186	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ( ( 1r ` R ) ( .r ` R ) x ) = x <-> ( ( 1r ` R ) ( .r ` S ) x ) = x ) )
19	16	oveqd 5891	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( x ( .r ` R ) ( 1r ` R ) ) = ( x ( .r ` S ) ( 1r ` R ) ) )
20	19	eqeq1d 2186	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ( x ( .r ` R ) ( 1r ` R ) ) = x <-> ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
21	18, 20	anbi12d 473	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ( ( ( 1r ` R ) ( .r ` R ) x ) = x /\ ( x ( .r ` R ) ( 1r ` R ) ) = x ) <-> ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) ) )
22	21	biimpa 296	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( ( ( 1r ` R ) ( .r ` R ) x ) = x /\ ( x ( .r ` R ) ( 1r ` R ) ) = x ) ) -> ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
23	14, 22	syldan 282	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` R ) ) -> ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
24	10, 23	syldan 282	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` S ) ) -> ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
25	24	ralrimiva 2550	. . 3 |- ( A e. (SubRing ` R ) -> A. x e. ( Base ` S ) ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
26	4	subrgring 13305	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
27		eqid 2177	. . . . 5 |- ( Base ` S ) = ( Base ` S )
28		eqid 2177	. . . . 5 |- ( .r ` S ) = ( .r ` S )
29		eqid 2177	. . . . 5 |- ( 1r ` S ) = ( 1r ` S )
30	27, 28, 29	isringid 13161	. . . 4 |- ( S e. Ring -> ( ( ( 1r ` R ) e. ( Base ` S ) /\ A. x e. ( Base ` S ) ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) ) <-> ( 1r ` S ) = ( 1r ` R ) ) )
31	26, 30	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> ( ( ( 1r ` R ) e. ( Base ` S ) /\ A. x e. ( Base ` S ) ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) ) <-> ( 1r ` S ) = ( 1r ` R ) ) )
32	6, 25, 31	mpbi2and 943	. 2 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` R ) )
33	1, 32	eqtr4id 2229	1 |- ( A e. (SubRing ` R ) -> .1. = ( 1r ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5216  (class class class)co 5874   Basecbs 12456   ↾_s cress 12457   .rcmulr 12531   1rcur 13095   Ringcrg 13132  SubRingcsubrg 13298

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-iress 12464  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-subg 12983  df-mgp 13084  df-ur 13096  df-ring 13134  df-subrg 13300

This theorem is referenced by:  subrguss  13317  subrginv  13318  subrgunit  13320  subrgnzr  13323  subsubrg  13326  zring1  13382