Theorem qusaddvallemg 13631

Description: Value of an operation defined on 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 )
qusaddflem.f	|- F = ( x e. V |-> [ x ] .~ )
qusaddflem.g	|- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
qusaddflemg.x	|- ( ph -> .x. e. W )

Assertion

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

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

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

Proof of Theorem qusaddvallemg

Step	Hyp	Ref	Expression
1		qusaddf.u	. . . 4 |- ( ph -> U = ( R /.s .~ ) )
2		qusaddf.v	. . . 4 |- ( ph -> V = ( Base ` R ) )
3		qusaddflem.f	. . . 4 |- F = ( x e. V |-> [ x ] .~ )
4		qusaddf.r	. . . . 5 |- ( ph -> .~ Er V )
5		qusaddf.z	. . . . . . 7 |- ( ph -> R e. Z )
6		basfn 13389	. . . . . . . 8 |- Base Fn _V
7		elex 2833	. . . . . . . 8 |- ( R e. Z -> R e. _V )
8		funfvex 5707	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5478	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	6, 7, 9	sylancr 418	. . . . . . 7 |- ( R e. Z -> ( Base ` R ) e. _V )
11	5, 10	syl 14	. . . . . 6 |- ( ph -> ( Base ` R ) e. _V )
12	2, 11	eqeltrd 2315	. . . . 5 |- ( ph -> V e. _V )
13		erex 6821	. . . . 5 |- ( .~ Er V -> ( V e. _V -> .~ e. _V ) )
14	4, 12, 13	sylc 62	. . . 4 |- ( ph -> .~ e. _V )
15	1, 2, 3, 14, 5	quslem 13622	. . 3 |- ( ph -> F : V -onto-> ( V /. .~ ) )
16		qusaddf.c	. . . 4 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
17		qusaddf.e	. . . 4 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
18	4, 12, 3, 16, 17	ercpbl 13629	. . 3 |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) )
19		qusaddflem.g	. . 3 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
20		qusaddflemg.x	. . 3 |- ( ph -> .x. e. W )
21	15, 18, 19, 12, 20	imasaddvallemg 13613	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )
22	4	3ad2ant1 1049	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> .~ Er V )
23	12	3ad2ant1 1049	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> V e. _V )
24		simp2 1029	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> X e. V )
25	22, 23, 3, 24	divsfvalg 13627	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` X ) = [ X ] .~ )
26		simp3 1030	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> Y e. V )
27	22, 23, 3, 26	divsfvalg 13627	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` Y ) = [ Y ] .~ )
28	25, 27	oveq12d 6093	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( [ X ] .~ .xb [ Y ] .~ ) )
29	16	3ad2antl1 1190	. . . 4 |- ( ( ( ph /\ X e. V /\ Y e. V ) /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
30	29, 24, 26	caovcld 6233	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( X .x. Y ) e. V )
31	22, 23, 3, 30	divsfvalg 13627	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` ( X .x. Y ) ) = [ ( X .x. Y ) ] .~ )
32	21, 28, 31	3eqtr3d 2279	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3705   <.cop 3708   U_ciun 4007   class class class wbr 4125   |-> cmpt 4187   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Er wer 6794   [cec 6795   /.cqs 6796   Basecbs 13330   /._s cqus 13600

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-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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-er 6797  df-ec 6799  df-qs 6803  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336

This theorem is referenced by:  qusaddval  13633  qusmulval  13635