Theorem qusaddvallemg 12752

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 12520	. . . . . . . 8 |- Base Fn _V
7		elex 2749	. . . . . . . 8 |- ( R e. Z -> R e. _V )
8		funfvex 5533	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5317	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	6, 7, 9	sylancr 414	. . . . . . 7 |- ( R e. Z -> ( Base ` R ) e. _V )
11	5, 10	syl 14	. . . . . 6 |- ( ph -> ( Base ` R ) e. _V )
12	2, 11	eqeltrd 2254	. . . . 5 |- ( ph -> V e. _V )
13		erex 6559	. . . . 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 12745	. . 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 12750	. . 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 12736	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )
22	4	3ad2ant1 1018	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> .~ Er V )
23	12	3ad2ant1 1018	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> V e. _V )
24		simp2 998	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> X e. V )
25	22, 23, 3, 24	divsfvalg 12748	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` X ) = [ X ] .~ )
26		simp3 999	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> Y e. V )
27	22, 23, 3, 26	divsfvalg 12748	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` Y ) = [ Y ] .~ )
28	25, 27	oveq12d 5893	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( [ X ] .~ .xb [ Y ] .~ ) )
29	16	3ad2antl1 1159	. . . 4 |- ( ( ( ph /\ X e. V /\ Y e. V ) /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
30	29, 24, 26	caovcld 6028	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( X .x. Y ) e. V )
31	22, 23, 3, 30	divsfvalg 12748	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` ( X .x. Y ) ) = [ ( X .x. Y ) ] .~ )
32	21, 28, 31	3eqtr3d 2218	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   _Vcvv 2738   {csn 3593   <.cop 3596   U_ciun 3887   class class class wbr 4004   |-> cmpt 4065   Fn wfn 5212   ` cfv 5217  (class class class)co 5875   Er wer 6532   [cec 6533   /.cqs 6534   Basecbs 12462   /._s cqus 12721

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-er 6535  df-ec 6537  df-qs 6541  df-inn 8920  df-ndx 12465  df-slot 12466  df-base 12468

This theorem is referenced by:  qusaddval  12754  qusmulval  12756