Theorem srgidmlem 13166

Description: Lemma for srglidm 13167 and srgridm 13168. (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 13156	. . 3 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
3		srgidm.b	. . . . . 6 |- B = ( Base ` R )
4	1, 3	mgpbasg 13141	. . . . 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 12840	. . 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 13139	. . . . . 6 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
14		srgidm.u	. . . . . . 7 |- .1. = ( 1r ` R )
15	1, 14	ringidvalg 13149	. . . . . 6 |- ( R e. SRing -> .1. = ( 0g ` (mulGrp ` R ) ) )
16		eqidd 2178	. . . . . 6 |- ( R e. SRing -> X = X )
17	13, 15, 16	oveq123d 5898	. . . . 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 5898	. . . . 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 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   .rcmulr 12539   0gc0g 12710   Mndcmnd 12822  mulGrpcmgp 13135   1rcur 13147  SRingcsrg 13151

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-mgp 13136  df-ur 13148  df-srg 13152

This theorem is referenced by:  srglidm  13167  srgridm  13168