Theorem subrg1 14539

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 2238	. . . . 5 |- ( 1r ` R ) = ( 1r ` R )
3	2	subrg1cl 14537	. . . 4 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
4		subrg1.1	. . . . 5 |- S = ( R ↾s A )
5	4	subrgbas 14538	. . . 4 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
6	3, 5	eleqtrd 2317	. . 3 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. ( Base ` S ) )
7		eqid 2238	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
8	7	subrgss 14530	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
9	5, 8	eqsstrrd 3285	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
10	9	sselda 3248	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` S ) ) -> x e. ( Base ` R ) )
11		subrgrcl 14534	. . . . . . 7 |- ( A e. (SubRing ` R ) -> R e. Ring )
12		eqid 2238	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
13	7, 12, 2	ringidmlem 14327	. . . . . . 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 13499	. . . . . . . . . . 11 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
16	11, 15	mpdan 425	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
17	16	oveqd 6102	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( ( 1r ` R ) ( .r ` R ) x ) = ( ( 1r ` R ) ( .r ` S ) x ) )
18	17	eqeq1d 2247	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ( ( 1r ` R ) ( .r ` R ) x ) = x <-> ( ( 1r ` R ) ( .r ` S ) x ) = x ) )
19	16	oveqd 6102	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( x ( .r ` R ) ( 1r ` R ) ) = ( x ( .r ` S ) ( 1r ` R ) ) )
20	19	eqeq1d 2247	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ( x ( .r ` R ) ( 1r ` R ) ) = x <-> ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
21	18, 20	anbi12d 477	. . . . . . 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 2623	. . 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 14532	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
27		eqid 2238	. . . . 5 |- ( Base ` S ) = ( Base ` S )
28		eqid 2238	. . . . 5 |- ( .r ` S ) = ( .r ` S )
29		eqid 2238	. . . . 5 |- ( 1r ` S ) = ( 1r ` S )
30	27, 28, 29	isringid 14330	. . . 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 956	. 2 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` R ) )
33	1, 32	eqtr4id 2290	1 |- ( A e. (SubRing ` R ) -> .1. = ( 1r ` S ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   Basecbs 13352   ↾_s cress 13353   .rcmulr 13432   1rcur 14262   Ringcrg 14300  SubRingcsubrg 14525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-subg 13973  df-mgp 14218  df-ur 14263  df-ring 14302  df-subrg 14527

This theorem is used by:  subrguss  14544  subrginv  14545  subrgunit  14547  subrgnzr  14550  subsubrg  14553  sralmod  14787  zring1  14936  ressascl  15039