Theorem qusmulrng 14869

Description: Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14870. Similar to qusmul2 14866. (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 14027	. . . 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 14860	. . 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 14243	. . . 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 13658	. 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 5377  (class class class)co 6085   Er wer 6804   [cec 6805   Basecbs 13352   .rcmulr 13432   /._s cqus 13623  SubGrpcsubg 13970   ~_QG cqg 13972  Rngcrng 14231  2Idealc2idl 14836

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

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 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-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-tpos 6516  df-er 6807  df-ec 6809  df-qs 6813  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-sca 13447  df-vsca 13448  df-ip 13449  df-0g 13612  df-iimas 13624  df-qus 13625  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-sbg 13810  df-subg 13973  df-eqg 13975  df-cmn 14089  df-abl 14090  df-mgp 14218  df-rng 14232  df-oppr 14373  df-lssm 14690  df-sra 14772  df-rgmod 14773  df-lidl 14806  df-2idl 14837

This theorem is used by:  quscrng  14870