Theorem qusmulrng 13871

Description: Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 13872. Similar to qusmul2 13868. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)

Hypotheses

Ref	Expression
qusmulrng.e	|- .~ = ( R ~QG S )
qusmulrng.h	|- H = ( R /.s .~ )
qusmulrng.b	|- B = ( Base ` R )
qusmulrng.p	|- .x. = ( .r ` R )
qusmulrng.a	|- .xb = ( .r ` H )

Assertion

Ref	Expression
qusmulrng	|- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( X e. B /\ Y e. B ) ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Proof of Theorem qusmulrng

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusmulrng.h	. . . 4 |- H = ( R /.s .~ )
2	1	a1i 9	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> H = ( R /.s .~ ) )
3		qusmulrng.b	. . . 4 |- B = ( Base ` R )
4	3	a1i 9	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> B = ( Base ` R ) )
5		qusmulrng.e	. . . . 5 |- .~ = ( R ~QG S )
6	3, 5	eqger 13188	. . . 4 |- ( S e. (SubGrp ` R ) -> .~ Er B )
7	6	3ad2ant3 1022	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> .~ Er B )
8		simp1 999	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> R e. Rng )
9		eqid 2189	. . . 4 |- (2Ideal ` R ) = (2Ideal ` R )
10		qusmulrng.p	. . . 4 |- .x. = ( .r ` R )
11	3, 5, 9, 10	2idlcpblrng 13863	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> ( ( a .~ b /\ c .~ d ) -> ( a .x. c ) .~ ( b .x. d ) ) )
12	8	anim1i 340	. . . . 5 |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( b e. B /\ d e. B ) ) -> ( R e. Rng /\ ( b e. B /\ d e. B ) ) )
13		3anass 984	. . . . 5 |- ( ( R e. Rng /\ b e. B /\ d e. B ) <-> ( R e. Rng /\ ( b e. B /\ d e. B ) ) )
14	12, 13	sylibr 134	. . . 4 |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( b e. B /\ d e. B ) ) -> ( R e. Rng /\ b e. B /\ d e. B ) )
15	3, 10	rngcl 13323	. . . 4 |- ( ( R e. Rng /\ b e. B /\ d e. B ) -> ( b .x. d ) e. B )
16	14, 15	syl 14	. . 3 |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( b e. B /\ d e. B ) ) -> ( b .x. d ) e. B )
17		qusmulrng.a	. . 3 |- .xb = ( .r ` H )
18	2, 4, 7, 8, 11, 16, 10, 17	qusmulval 12824	. 2 |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ X e. B /\ Y e. B ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )
19	18	3expb 1206	1 |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( X e. B /\ Y e. B ) ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   ` cfv 5238  (class class class)co 5900   Er wer 6560   [cec 6561   Basecbs 12523   .rcmulr 12601   /._s cqus 12788  SubGrpcsubg 13131   ~_QG cqg 13133  Rngcrng 13311  2Idealc2idl 13840

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-pre-ltirr 7958  ax-pre-lttrn 7960  ax-pre-ltadd 7962

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-tp 3618  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-tpos 6274  df-er 6563  df-ec 6565  df-qs 6569  df-pnf 8029  df-mnf 8030  df-ltxr 8032  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-5 9016  df-6 9017  df-7 9018  df-8 9019  df-ndx 12526  df-slot 12527  df-base 12529  df-sets 12530  df-iress 12531  df-plusg 12613  df-mulr 12614  df-sca 12616  df-vsca 12617  df-ip 12618  df-0g 12774  df-iimas 12790  df-qus 12791  df-mgm 12843  df-sgrp 12888  df-mnd 12901  df-grp 12971  df-minusg 12972  df-sbg 12973  df-subg 13134  df-eqg 13136  df-cmn 13250  df-abl 13251  df-mgp 13300  df-rng 13312  df-oppr 13443  df-lssm 13694  df-sra 13776  df-rgmod 13777  df-lidl 13810  df-2idl 13841

This theorem is referenced by:  quscrng  13872