Theorem srgidmlem 13978

Description: Lemma for srglidm 13979 and srgridm 13980. (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 2229	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 13968	. . 3 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
3		srgidm.b	. . . . . 6 |- B = ( Base ` R )
4	1, 3	mgpbasg 13926	. . . . 5 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
5	4	eleq2d 2299	. . . 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 2229	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
8		eqid 2229	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
9		eqid 2229	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
10	7, 8, 9	mndlrid 13504	. . 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 597	. 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 13924	. . . . . 6 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
14		srgidm.u	. . . . . . 7 |- .1. = ( 1r ` R )
15	1, 14	ringidvalg 13961	. . . . . 6 |- ( R e. SRing -> .1. = ( 0g ` (mulGrp ` R ) ) )
16		eqidd 2230	. . . . . 6 |- ( R e. SRing -> X = X )
17	13, 15, 16	oveq123d 6032	. . . . 5 |- ( R e. SRing -> ( .1. .x. X ) = ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) )
18	17	eqeq1d 2238	. . . 4 |- ( R e. SRing -> ( ( .1. .x. X ) = X <-> ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X ) )
19	13, 16, 15	oveq123d 6032	. . . . 5 |- ( R e. SRing -> ( X .x. .1. ) = ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) )
20	19	eqeq1d 2238	. . . 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 1395   e. wcel 2200   ` cfv 5322  (class class class)co 6011   Basecbs 13069   +g cplusg 13147   .rcmulr 13148   0gc0g 13326   Mndcmnd 13486  mulGrpcmgp 13920   1rcur 13959  SRingcsrg 13963

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-pre-ltirr 8132  ax-pre-ltadd 8136

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-pnf 8204  df-mnf 8205  df-ltxr 8207  df-inn 9132  df-2 9190  df-3 9191  df-ndx 13072  df-slot 13073  df-base 13075  df-sets 13076  df-plusg 13160  df-mulr 13161  df-0g 13328  df-mgm 13426  df-sgrp 13472  df-mnd 13487  df-mgp 13921  df-ur 13960  df-srg 13964

This theorem is referenced by:  srglidm  13979  srgridm  13980