Theorem subrg1 16967

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		eqid 2450	. . . . 5 |- ( 1r ` R ) = ( 1r ` R )
2	1	subrg1cl 16965	. . . 4 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
3		subrg1.1	. . . . 5 |- S = ( R ↾s A )
4	3	subrgbas 16966	. . . 4 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
5	2, 4	eleqtrd 2538	. . 3 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. ( Base ` S ) )
6		eqid 2450	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
7	6	subrgss 16958	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
8	4, 7	eqsstr3d 3475	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
9	8	sselda 3440	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` S ) ) -> x e. ( Base ` R ) )
10		subrgrcl 16962	. . . . . . 7 |- ( A e. (SubRing ` R ) -> R e. Ring )
11		eqid 2450	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
12	6, 11, 1	rngidmlem 16759	. . . . . . 7 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( ( 1r ` R ) ( .r ` R ) x ) = x /\ ( x ( .r ` R ) ( 1r ` R ) ) = x ) )
13	10, 12	sylan 471	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` R ) ) -> ( ( ( 1r ` R ) ( .r ` R ) x ) = x /\ ( x ( .r ` R ) ( 1r ` R ) ) = x ) )
14	3, 11	ressmulr 14379	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
15	14	oveqd 6193	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( ( 1r ` R ) ( .r ` R ) x ) = ( ( 1r ` R ) ( .r ` S ) x ) )
16	15	eqeq1d 2452	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ( ( 1r ` R ) ( .r ` R ) x ) = x <-> ( ( 1r ` R ) ( .r ` S ) x ) = x ) )
17	14	oveqd 6193	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( x ( .r ` R ) ( 1r ` R ) ) = ( x ( .r ` S ) ( 1r ` R ) ) )
18	17	eqeq1d 2452	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ( x ( .r ` R ) ( 1r ` R ) ) = x <-> ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
19	16, 18	anbi12d 710	. . . . . . 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 ) ) )
20	19	biimpa 484	. . . . . 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 ) )
21	13, 20	syldan 470	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` R ) ) -> ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
22	9, 21	syldan 470	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. ( Base ` S ) ) -> ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
23	22	ralrimiva 2881	. . 3 |- ( A e. (SubRing ` R ) -> A. x e. ( Base ` S ) ( ( ( 1r ` R ) ( .r ` S ) x ) = x /\ ( x ( .r ` S ) ( 1r ` R ) ) = x ) )
24	3	subrgrng 16960	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
25		eqid 2450	. . . . 5 |- ( Base ` S ) = ( Base ` S )
26		eqid 2450	. . . . 5 |- ( .r ` S ) = ( .r ` S )
27		eqid 2450	. . . . 5 |- ( 1r ` S ) = ( 1r ` S )
28	25, 26, 27	isrngid 16762	. . . 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 ) ) )
29	24, 28	syl 16	. . 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 ) ) )
30	5, 23, 29	mpbi2and 912	. 2 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` R ) )
31		subrg1.2	. 2 |- .1. = ( 1r ` R )
32	30, 31	syl6reqr 2509	1 |- ( A e. (SubRing ` R ) -> .1. = ( 1r ` S ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1757   A.wral 2792   ` cfv 5502  (class class class)co 6176   Basecbs 14262   ↾_s cress 14263   .rcmulr 14327   1rcur 16694   Ringcrg 16737  SubRingcsubrg 16953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-mulr 14340  df-0g 14468  df-mnd 15503  df-subg 15766  df-mgp 16683  df-ur 16695  df-rng 16739  df-subrg 16955

This theorem is referenced by:  subrguss  16972  subrginv  16973  subrgunit  16975  subsubrg  16983  sralmod  17360  subrgnzr  17441  ressascl  17506  mpl1  17616  subrgmvr  17633  gzrngunitlem  17972  zring1  17989  zrng1  17995  prmirredlemOLD  18015  mulgrhmOLD  18024  mulgrhm2OLD  18025  re1r  18138  clm1  20747  qrng1  22973  subrgchr  26382  scmatsrng1  31030