Theorem qusaddvallem 16211

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 ) ) >. } )

Assertion

Ref	Expression
qusaddvallem	|- ( ( 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)   X( a, b)   Y( a, b)   Z( x, q, p, a, b)

Proof of Theorem qusaddvallem

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		fvex 6201	. . . . . 6 |- ( Base ` R ) e. _V
6	2, 5	syl6eqel 2709	. . . . 5 |- ( ph -> V e. _V )
7		erex 7766	. . . . 5 |- ( .~ Er V -> ( V e. _V -> .~ e. _V ) )
8	4, 6, 7	sylc 65	. . . 4 |- ( ph -> .~ e. _V )
9		qusaddf.z	. . . 4 |- ( ph -> R e. Z )
10	1, 2, 3, 8, 9	quslem 16203	. . 3 |- ( ph -> F : V -onto-> ( V /. .~ ) )
11		qusaddf.c	. . . 4 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
12		qusaddf.e	. . . 4 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
13	4, 6, 3, 11, 12	ercpbl 16209	. . 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 ) ) ) )
14		qusaddflem.g	. . 3 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
15	10, 13, 14	imasaddvallem 16189	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )
16	4	3ad2ant1 1082	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> .~ Er V )
17	6	3ad2ant1 1082	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> V e. _V )
18	16, 17, 3	divsfval 16207	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` X ) = [ X ] .~ )
19	16, 17, 3	divsfval 16207	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` Y ) = [ Y ] .~ )
20	18, 19	oveq12d 6668	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( [ X ] .~ .xb [ Y ] .~ ) )
21	16, 17, 3	divsfval 16207	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` ( X .x. Y ) ) = [ ( X .x. Y ) ] .~ )
22	15, 20, 21	3eqtr3d 2664	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Er wer 7739   [cec 7740   /.cqs 7741   Basecbs 15857   /._s cqus 16165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-er 7742  df-ec 7744  df-qs 7748

This theorem is referenced by:  qusaddval  16213  qusmulval  16215