Theorem srgidmlem 13114

Description: Lemma for srglidm 13115 and srgridm 13116. (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 2177	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 13104	. . 3 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
3		srgidm.b	. . . . . 6 |- B = ( Base ` R )
4	1, 3	mgpbasg 13089	. . . . 5 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
5	4	eleq2d 2247	. . . 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 2177	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
8		eqid 2177	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
9		eqid 2177	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
10	7, 8, 9	mndlrid 12789	. . 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 13087	. . . . . 6 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
14		srgidm.u	. . . . . . 7 |- .1. = ( 1r ` R )
15	1, 14	ringidvalg 13097	. . . . . 6 |- ( R e. SRing -> .1. = ( 0g ` (mulGrp ` R ) ) )
16		eqidd 2178	. . . . . 6 |- ( R e. SRing -> X = X )
17	13, 15, 16	oveq123d 5895	. . . . 5 |- ( R e. SRing -> ( .1. .x. X ) = ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) )
18	17	eqeq1d 2186	. . . 4 |- ( R e. SRing -> ( ( .1. .x. X ) = X <-> ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X ) )
19	13, 16, 15	oveq123d 5895	. . . . 5 |- ( R e. SRing -> ( X .x. .1. ) = ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) )
20	19	eqeq1d 2186	. . . 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 1353   e. wcel 2148   ` cfv 5216  (class class class)co 5874   Basecbs 12456   +g cplusg 12530   .rcmulr 12531   0gc0g 12695   Mndcmnd 12771  mulGrpcmgp 13083   1rcur 13095  SRingcsrg 13099

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-mgp 13084  df-ur 13096  df-srg 13100

This theorem is referenced by:  srglidm  13115  srgridm  13116