Theorem ringidmlem 13899

Description: Lemma for ringlidm 13900 and ringridm 13901. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Hypotheses

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

Assertion

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

Proof of Theorem ringidmlem

Step	Hyp	Ref	Expression
1		eqid 2207	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	ringmgp 13879	. . 3 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
3		simpr 110	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> X e. B )
4		rngidm.b	. . . . . 6 |- B = ( Base ` R )
5	1, 4	mgpbasg 13803	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
6	5	adantr 276	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> B = ( Base ` (mulGrp ` R ) ) )
7	3, 6	eleqtrd 2286	. . 3 |- ( ( R e. Ring /\ X e. B ) -> X e. ( Base ` (mulGrp ` R ) ) )
8		eqid 2207	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
9		eqid 2207	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
10		eqid 2207	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
11	8, 9, 10	mndlrid 13381	. . 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 ) )
12	2, 7, 11	syl2an2r 595	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X /\ ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) = X ) )
13		rngidm.t	. . . . . . 7 |- .x. = ( .r ` R )
14	1, 13	mgpplusgg 13801	. . . . . 6 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
15	14	adantr 276	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> .x. = ( +g ` (mulGrp ` R ) ) )
16		rngidm.u	. . . . . . 7 |- .1. = ( 1r ` R )
17	1, 16	ringidvalg 13838	. . . . . 6 |- ( R e. Ring -> .1. = ( 0g ` (mulGrp ` R ) ) )
18	17	adantr 276	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> .1. = ( 0g ` (mulGrp ` R ) ) )
19		eqidd 2208	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> X = X )
20	15, 18, 19	oveq123d 5988	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) )
21	20	eqeq1d 2216	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( ( .1. .x. X ) = X <-> ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X ) )
22	15, 19, 18	oveq123d 5988	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) )
23	22	eqeq1d 2216	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( ( X .x. .1. ) = X <-> ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) = X ) )
24	21, 23	anbi12d 473	. 2 |- ( ( R e. Ring /\ 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 ) ) )
25	12, 24	mpbird 167	1 |- ( ( R e. Ring /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   .rcmulr 13025   0gc0g 13203   Mndcmnd 13363  mulGrpcmgp 13797   1rcur 13836   Ringcrg 13873

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-mgp 13798  df-ur 13837  df-ring 13875

This theorem is referenced by:  ringlidm  13900  ringridm  13901  ringid  13903  subrg1  14108