Theorem qusmulrng 14028

Description: Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14029. Similar to qusmul2 14025. (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 13294	. . . 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 2193	. . . 4 |- (2Ideal ` R ) = (2Ideal ` R )
10		qusmulrng.p	. . . 4 |- .x. = ( .r ` R )
11	3, 5, 9, 10	2idlcpblrng 14019	. . 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 13440	. . . 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 12920	. 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 2164   ` cfv 5254  (class class class)co 5918   Er wer 6584   [cec 6585   Basecbs 12618   .rcmulr 12696   /._s cqus 12883  SubGrpcsubg 13237   ~_QG cqg 13239  Rngcrng 13428  2Idealc2idl 13995

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-tpos 6298  df-er 6587  df-ec 6589  df-qs 6593  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-0g 12869  df-iimas 12885  df-qus 12886  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-subg 13240  df-eqg 13242  df-cmn 13356  df-abl 13357  df-mgp 13417  df-rng 13429  df-oppr 13564  df-lssm 13849  df-sra 13931  df-rgmod 13932  df-lidl 13965  df-2idl 13996

This theorem is referenced by:  quscrng  14029