Theorem qusaddvallemg 13546

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 13271	. . . . . . . 8 |- Base Fn _V
7		elex 2825	. . . . . . . 8 |- ( R e. Z -> R e. _V )
8		funfvex 5687	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5458	. . . . . . . 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 2309	. . . . 5 |- ( ph -> V e. _V )
13		erex 6791	. . . . 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 13537	. . 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 13544	. . 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 13528	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )
22	4	3ad2ant1 1045	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> .~ Er V )
23	12	3ad2ant1 1045	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> V e. _V )
24		simp2 1025	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> X e. V )
25	22, 23, 3, 24	divsfvalg 13542	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` X ) = [ X ] .~ )
26		simp3 1026	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> Y e. V )
27	22, 23, 3, 26	divsfvalg 13542	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` Y ) = [ Y ] .~ )
28	25, 27	oveq12d 6068	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( [ X ] .~ .xb [ Y ] .~ ) )
29	16	3ad2antl1 1186	. . . 4 |- ( ( ( ph /\ X e. V /\ Y e. V ) /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
30	29, 24, 26	caovcld 6208	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( X .x. Y ) e. V )
31	22, 23, 3, 30	divsfvalg 13542	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` ( X .x. Y ) ) = [ ( X .x. Y ) ] .~ )
32	21, 28, 31	3eqtr3d 2273	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   _Vcvv 2813   {csn 3689   <.cop 3692   U_ciun 3991   class class class wbr 4109   |-> cmpt 4171   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Er wer 6764   [cec 6765   /.cqs 6766   Basecbs 13212   /._s cqus 13513

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-er 6767  df-ec 6769  df-qs 6773  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218

This theorem is referenced by:  qusaddval  13548  qusmulval  13550