Theorem qusmulrng 14978

Description: Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14979. Similar to qusmul2 14975. (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 14086	. . . 4 |- ( S e. (SubGrp ` R ) -> .~ Er B )
7	6	3ad2ant3 1051	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> .~ Er B )
8		simp1 1028	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> R e. Rng )
9		eqid 2238	. . . 4 |- (2Ideal ` R ) = (2Ideal ` R )
10		qusmulrng.p	. . . 4 |- .x. = ( .r ` R )
11	3, 5, 9, 10	2idlcpblrng 14969	. . 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 1013	. . . . 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 14350	. . . 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 13717	. 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 1235	1 |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( X e. B /\ Y e. B ) ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   Er wer 6808   [cec 6809   Basecbs 13410   .rcmulr 13491   /._s cqus 13682  SubGrpcsubg 14029   ~_QG cqg 14031  Rngcrng 14338  2Idealc2idl 14945

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-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3or 1010  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-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-tpos 6520  df-er 6811  df-ec 6813  df-qs 6817  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-sca 13506  df-vsca 13507  df-ip 13508  df-0g 13671  df-iimas 13683  df-qus 13684  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-sbg 13869  df-subg 14032  df-eqg 14034  df-cmn 14179  df-abl 14180  df-mgp 14325  df-rng 14339  df-oppr 14482  df-lssm 14799  df-sra 14881  df-rgmod 14882  df-lidl 14915  df-2idl 14946

This theorem is used by:  quscrng  14979