Theorem qusaddvallemg 12802

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 12565	. . . . . . . 8 |- Base Fn _V
7		elex 2763	. . . . . . . 8 |- ( R e. Z -> R e. _V )
8		funfvex 5548	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5332	. . . . . . . 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 2266	. . . . 5 |- ( ph -> V e. _V )
13		erex 6578	. . . . 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 12794	. . 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 12800	. . 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 12785	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )
22	4	3ad2ant1 1020	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> .~ Er V )
23	12	3ad2ant1 1020	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> V e. _V )
24		simp2 1000	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> X e. V )
25	22, 23, 3, 24	divsfvalg 12798	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` X ) = [ X ] .~ )
26		simp3 1001	. . . 4 |- ( ( ph /\ X e. V /\ Y e. V ) -> Y e. V )
27	22, 23, 3, 26	divsfvalg 12798	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` Y ) = [ Y ] .~ )
28	25, 27	oveq12d 5910	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( [ X ] .~ .xb [ Y ] .~ ) )
29	16	3ad2antl1 1161	. . . 4 |- ( ( ( ph /\ X e. V /\ Y e. V ) /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
30	29, 24, 26	caovcld 6046	. . 3 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( X .x. Y ) e. V )
31	22, 23, 3, 30	divsfvalg 12798	. 2 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( F ` ( X .x. Y ) ) = [ ( X .x. Y ) ] .~ )
32	21, 28, 31	3eqtr3d 2230	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   _Vcvv 2752   {csn 3607   <.cop 3610   U_ciun 3901   class class class wbr 4018   |-> cmpt 4079   Fn wfn 5227   ` cfv 5232  (class class class)co 5892   Er wer 6551   [cec 6552   /.cqs 6553   Basecbs 12507   /._s cqus 12770

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7927  ax-resscn 7928  ax-1re 7930  ax-addrcl 7933

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-er 6554  df-ec 6556  df-qs 6560  df-inn 8945  df-ndx 12510  df-slot 12511  df-base 12513

This theorem is referenced by:  qusaddval  12804  qusmulval  12806