Theorem subrg1 14326

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 14324	. . . 4 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
4		subrg1.1	. . . . 5 |- S = ( R ↾s A )
5	4	subrgbas 14325	. . . 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 14317	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
9	5, 8	eqsstrrd 3265	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
10	9	sselda 3228	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` S ) ) -> x e. ( Base ` R ) )
11		subrgrcl 14321	. . . . . . 7 |- ( A e. (SubRing ` R ) -> R e. Ring )
12		eqid 2231	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
13	7, 12, 2	ringidmlem 14116	. . . . . . 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 13308	. . . . . . . . . . 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 6045	. . . . . . . . 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 6045	. . . . . . . . 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 2606	. . 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 14319	. . . 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 14119	. . . 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 952	. 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 1398   e. wcel 2202   A.wral 2511   ` cfv 5333  (class class class)co 6028   Basecbs 13162   ↾_s cress 13163   .rcmulr 13241   1rcur 14053   Ringcrg 14090  SubRingcsubrg 14312

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-subg 13837  df-mgp 14015  df-ur 14054  df-ring 14092  df-subrg 14314

This theorem is referenced by:  subrguss  14331  subrginv  14332  subrgunit  14334  subrgnzr  14337  subsubrg  14340  sralmod  14546  zring1  14697