Theorem qusaddflemg 12807

Description: The operation of a quotient structure is a function. (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
qusaddflemg	|- ( ph -> .xb : ( ( V /. .~ ) X. ( V /. .~ ) ) --> ( V /. .~ ) )

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   .xb , a, b, p, q

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)   Z( x, q, p, a, b)

Proof of Theorem qusaddflemg

Step	Hyp	Ref	Expression
1		qusaddf.u	. . 3 |- ( ph -> U = ( R /.s .~ ) )
2		qusaddf.v	. . 3 |- ( ph -> V = ( Base ` R ) )
3		qusaddflem.f	. . 3 |- F = ( x e. V |-> [ x ] .~ )
4		qusaddf.r	. . . 4 |- ( ph -> .~ Er V )
5		basfn 12569	. . . . . 6 |- Base Fn _V
6		qusaddf.z	. . . . . . 7 |- ( ph -> R e. Z )
7	6	elexd 2765	. . . . . 6 |- ( ph -> R e. _V )
8		funfvex 5551	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5335	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	5, 7, 9	sylancr 414	. . . . 5 |- ( ph -> ( Base ` R ) e. _V )
11	2, 10	eqeltrd 2266	. . . 4 |- ( ph -> V e. _V )
12		erex 6582	. . . 4 |- ( .~ Er V -> ( V e. _V -> .~ e. _V ) )
13	4, 11, 12	sylc 62	. . 3 |- ( ph -> .~ e. _V )
14	1, 2, 3, 13, 6	quslem 12798	. 2 |- ( ph -> F : V -onto-> ( V /. .~ ) )
15		qusaddf.c	. . 3 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
16		qusaddf.e	. . 3 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
17	4, 11, 3, 15, 16	ercpbl 12804	. 2 |- ( ( 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 ) ) ) )
18		qusaddflem.g	. 2 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
19		qusaddflemg.x	. 2 |- ( ph -> .x. e. W )
20	14, 17, 18, 11, 19, 15	imasaddflemg 12790	1 |- ( ph -> .xb : ( ( V /. .~ ) X. ( V /. .~ ) ) --> ( V /. .~ ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   {csn 3607   <.cop 3610   U_ciun 3901   class class class wbr 4018   |-> cmpt 4079   X. cxp 4642   Fn wfn 5230   -->wf 5231   ` cfv 5235  (class class class)co 5895   Er wer 6555   [cec 6556   /.cqs 6557   Basecbs 12511   /._s cqus 12774

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 4192  ax-pr 4227  ax-un 4451  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

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 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-er 6558  df-ec 6560  df-qs 6564  df-inn 8949  df-ndx 12514  df-slot 12515  df-base 12517

This theorem is referenced by:  qusaddf  12809  qusmulf  12811