Theorem ringidmlem 14411

Description: Lemma for ringlidm 14412 and ringridm 14413. (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 2238	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	ringmgp 14390	. . 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 14308	. . . . 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 2317	. . 3 |- ( ( R e. Ring /\ X e. B ) -> X e. ( Base ` (mulGrp ` R ) ) )
8		eqid 2238	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
9		eqid 2238	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
10		eqid 2238	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
11	8, 9, 10	mndlrid 13800	. . 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 603	. 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 14305	. . . . . 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 14348	. . . . . 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 2239	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> X = X )
20	15, 18, 19	oveq123d 6106	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) )
21	20	eqeq1d 2247	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( ( .1. .x. X ) = X <-> ( ( 0g ` (mulGrp ` R ) ) ( +g ` (mulGrp ` R ) ) X ) = X ) )
22	15, 19, 18	oveq123d 6106	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) )
23	22	eqeq1d 2247	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( ( X .x. .1. ) = X <-> ( X ( +g ` (mulGrp ` R ) ) ( 0g ` (mulGrp ` R ) ) ) = X ) )
24	21, 23	anbi12d 477	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13404   +g cplusg 13484   .rcmulr 13485   0gc0g 13663   Mndcmnd 13782  mulGrpcmgp 14301   1rcur 14346   Ringcrg 14384

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-pre-ltirr 8292  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-ltxr 8366  df-inn 9308  df-2 9366  df-3 9367  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411  df-plusg 13497  df-mulr 13498  df-0g 13665  df-mgm 13729  df-sgrp 13770  df-mnd 13783  df-mgp 14302  df-ur 14347  df-ring 14386

This theorem is used by:  ringlidm  14412  ringridm  14413  ringid  14415  subrg1  14623