Theorem subrg1 14244

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 2231	. . . . 5 |- ( 1r ` R ) = ( 1r ` R )
3	2	subrg1cl 14242	. . . 4 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
4		subrg1.1	. . . . 5 |- S = ( R ↾s A )
5	4	subrgbas 14243	. . . 4 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
6	3, 5	eleqtrd 2310	. . 3 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. ( Base ` S ) )
7		eqid 2231	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
8	7	subrgss 14235	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
9	5, 8	eqsstrrd 3264	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
10	9	sselda 3227	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` S ) ) -> x e. ( Base ` R ) )
11		subrgrcl 14239	. . . . . . 7 |- ( A e. (SubRing ` R ) -> R e. Ring )
12		eqid 2231	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
13	7, 12, 2	ringidmlem 14034	. . . . . . 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 13227	. . . . . . . . . . 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 6034	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( ( 1r ` R ) ( .r ` R ) x ) = ( ( 1r ` R ) ( .r ` S ) x ) )
18	17	eqeq1d 2240	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ( ( 1r ` R ) ( .r ` R ) x ) = x <-> ( ( 1r ` R ) ( .r ` S ) x ) = x ) )
19	16	oveqd 6034	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( x ( .r ` R ) ( 1r ` R ) ) = ( x ( .r ` S ) ( 1r ` R ) ) )
20	19	eqeq1d 2240	. . . . . . . 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 2605	. . 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 14237	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
27		eqid 2231	. . . . 5 |- ( Base ` S ) = ( Base ` S )
28		eqid 2231	. . . . 5 |- ( .r ` S ) = ( .r ` S )
29		eqid 2231	. . . . 5 |- ( 1r ` S ) = ( 1r ` S )
30	27, 28, 29	isringid 14037	. . . 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 951	. 2 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` R ) )
33	1, 32	eqtr4id 2283	1 |- ( A e. (SubRing ` R ) -> .1. = ( 1r ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   .rcmulr 13160   1rcur 13971   Ringcrg 14008  SubRingcsubrg 14230

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-subg 13756  df-mgp 13933  df-ur 13972  df-ring 14010  df-subrg 14232

This theorem is referenced by:  subrguss  14249  subrginv  14250  subrgunit  14252  subrgnzr  14255  subsubrg  14258  sralmod  14463  zring1  14614