Theorem qusmulval 13111

Description: The multiplication in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
qusaddf.u	|- ( ph -> U = ( R /.s .~ ) )
qusaddf.v	|- ( ph -> V = ( Base ` R ) )
qusaddf.r	|- ( ph -> .~ Er V )
qusaddf.z	|- ( ph -> R e. Z )
qusaddf.e	|- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
qusaddf.c	|- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
qusmulf.p	|- .x. = ( .r ` R )
qusmulf.a	|- .xb = ( .r ` U )

Assertion

Ref	Expression
qusmulval	|- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Distinct variable groups:   a, b, p, q, .~   ph, a, b, p, q   V, a, b, p, q   R, p, q   .x. , p, q   X, p, q   .xb , a, b, p, q   Y, p, q

Allowed substitution hints:   R( a, b)   .x. ( a, b)   U( q, p, a, b)   X( a, b)   Y( a, b)   Z( q, p, a, b)

Proof of Theorem qusmulval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusaddf.u	. 2 |- ( ph -> U = ( R /.s .~ ) )
2		qusaddf.v	. 2 |- ( ph -> V = ( Base ` R ) )
3		qusaddf.r	. 2 |- ( ph -> .~ Er V )
4		qusaddf.z	. 2 |- ( ph -> R e. Z )
5		qusaddf.e	. 2 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
6		qusaddf.c	. 2 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
7		eqid 2204	. 2 |- ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] .~ )
8		basfn 12832	. . . . . . 7 |- Base Fn _V
9	4	elexd 2784	. . . . . . 7 |- ( ph -> R e. _V )
10		funfvex 5592	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
11	10	funfni 5375	. . . . . . 7 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
12	8, 9, 11	sylancr 414	. . . . . 6 |- ( ph -> ( Base ` R ) e. _V )
13	2, 12	eqeltrd 2281	. . . . 5 |- ( ph -> V e. _V )
14		erex 6643	. . . . 5 |- ( .~ Er V -> ( V e. _V -> .~ e. _V ) )
15	3, 13, 14	sylc 62	. . . 4 |- ( ph -> .~ e. _V )
16	1, 2, 7, 15, 4	qusval 13097	. . 3 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
17	1, 2, 7, 15, 4	quslem 13098	. . 3 |- ( ph -> ( x e. V |-> [ x ] .~ ) : V -onto-> ( V /. .~ ) )
18		qusmulf.p	. . 3 |- .x. = ( .r ` R )
19		qusmulf.a	. . 3 |- .xb = ( .r ` U )
20	16, 2, 17, 4, 18, 19	imasmulr 13083	. 2 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( ( x e. V |-> [ x ] .~ ) ` p ) , ( ( x e. V |-> [ x ] .~ ) ` q ) >. , ( ( x e. V |-> [ x ] .~ ) ` ( p .x. q ) ) >. } )
21		mulrslid 12906	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
22	21	slotex 12801	. . . 4 |- ( R e. Z -> ( .r ` R ) e. _V )
23	4, 22	syl 14	. . 3 |- ( ph -> ( .r ` R ) e. _V )
24	18, 23	eqeltrid 2291	. 2 |- ( ph -> .x. e. _V )
25	1, 2, 3, 4, 5, 6, 7, 20, 24	qusaddvallemg 13107	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1372   e. wcel 2175   _Vcvv 2771   class class class wbr 4043   |-> cmpt 4104   Fn wfn 5265   ` cfv 5270  (class class class)co 5943   Er wer 6616   [cec 6617   /.cqs 6618   Basecbs 12774   .rcmulr 12852   /._s cqus 13074

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-tp 3640  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-er 6619  df-ec 6621  df-qs 6625  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12777  df-slot 12778  df-base 12780  df-plusg 12864  df-mulr 12865  df-iimas 13076  df-qus 13077

This theorem is referenced by:  qusrhm  14232  qusmul2  14233  qusmulrng  14236