Theorem srgidmlem 13474

Description: Lemma for srglidm 13475 and srgridm 13476. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
srgidm.b	|- B = ( Base ` R )
srgidm.t	|- .x. = ( .r ` R )
srgidm.u	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
srgidmlem	|- ( ( R e. SRing /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) )

Proof of Theorem srgidmlem

Step	Hyp	Ref	Expression
1		eqid 2193	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 13464	. . 3 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
3		srgidm.b	. . . . . 6 |- B = ( Base ` R )
4	1, 3	mgpbasg 13422	. . . . 5 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
5	4	eleq2d 2263	. . . 4 |- ( R e. SRing -> ( X e. B <-> X e. ( Base ` (mulGrp ` R ) ) ) )
6	5	biimpa 296	. . 3 |- ( ( R e. SRing /\ X e. B ) -> X e. ( Base ` (mulGrp ` R ) ) )
7		eqid 2193	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
8		eqid 2193	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
9		eqid 2193	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
10	7, 8, 9	mndlrid 13015	. . 3 |- ( ( (mulGrp ` R ) e. Mnd /\ X e. ( Base ` (mulGrp ` R ) ) ) -> ( ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X /\ ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) = X ) )
11	2, 6, 10	syl2an2r 595	. 2 |- ( ( R e. SRing /\ X e. B ) -> ( ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X /\ ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) = X ) )
12		srgidm.t	. . . . . . 7 |- .x. = ( .r ` R )
13	1, 12	mgpplusgg 13420	. . . . . 6 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
14		srgidm.u	. . . . . . 7 |- .1. = ( 1r ` R )
15	1, 14	ringidvalg 13457	. . . . . 6 |- ( R e. SRing -> .1. = ( 0g ` (mulGrp ` R ) ) )
16		eqidd 2194	. . . . . 6 |- ( R e. SRing -> X = X )
17	13, 15, 16	oveq123d 5939	. . . . 5 |- ( R e. SRing -> ( .1. .x. X ) = ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) )
18	17	eqeq1d 2202	. . . 4 |- ( R e. SRing -> ( ( .1. .x. X ) = X <-> ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X ) )
19	13, 16, 15	oveq123d 5939	. . . . 5 |- ( R e. SRing -> ( X .x. .1. ) = ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) )
20	19	eqeq1d 2202	. . . 4 |- ( R e. SRing -> ( ( X .x. .1. ) = X <-> ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) = X ) )
21	18, 20	anbi12d 473	. . 3 |- ( R e. SRing -> ( ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) <-> ( ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X /\ ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) = X ) ) )
22	21	adantr 276	. 2 |- ( ( R e. SRing /\ X e. B ) -> ( ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) <-> ( ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X /\ ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) = X ) ) )
23	11, 22	mpbird 167	1 |- ( ( R e. SRing /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   .rcmulr 12696   0gc0g 12867   Mndcmnd 12997  mulGrpcmgp 13416   1rcur 13455  SRingcsrg 13459

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mgp 13417  df-ur 13456  df-srg 13460

This theorem is referenced by:  srglidm  13475  srgridm  13476