Theorem qusmulrng 14369

Description: Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14370. Similar to qusmul2 14366. (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 13635	. . . 4 |- ( S e. (SubGrp ` R ) -> .~ Er B )
7	6	3ad2ant3 1023	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> .~ Er B )
8		simp1 1000	. . 3 |- ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) -> R e. Rng )
9		eqid 2206	. . . 4 |- (2Ideal ` R ) = (2Ideal ` R )
10		qusmulrng.p	. . . 4 |- .x. = ( .r ` R )
11	3, 5, 9, 10	2idlcpblrng 14360	. . 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 985	. . . . 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 13781	. . . 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 13244	. 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 1207	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 981   = wceq 1373   e. wcel 2177   ` cfv 5280  (class class class)co 5957   Er wer 6630   [cec 6631   Basecbs 12907   .rcmulr 12985   /._s cqus 13207  SubGrpcsubg 13578   ~_QG cqg 13580  Rngcrng 13769  2Idealc2idl 14336

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 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-tp 3646  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-tpos 6344  df-er 6633  df-ec 6635  df-qs 6639  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-iress 12915  df-plusg 12997  df-mulr 12998  df-sca 13000  df-vsca 13001  df-ip 13002  df-0g 13165  df-iimas 13209  df-qus 13210  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-sbg 13412  df-subg 13581  df-eqg 13583  df-cmn 13697  df-abl 13698  df-mgp 13758  df-rng 13770  df-oppr 13905  df-lssm 14190  df-sra 14272  df-rgmod 14273  df-lidl 14306  df-2idl 14337

This theorem is referenced by:  quscrng  14370