Theorem qusaddflemg 13598

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 13355	. . . . . 6 |- Base Fn _V
6		qusaddf.z	. . . . . . 7 |- ( ph -> R e. Z )
7	6	elexd 2829	. . . . . 6 |- ( ph -> R e. _V )
8		funfvex 5692	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5463	. . . . . 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 2311	. . . 4 |- ( ph -> V e. _V )
12		erex 6804	. . . 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 13588	. 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 13595	. 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 13580	1 |- ( ph -> .xb : ( ( V /. .~ ) X. ( V /. .~ ) ) --> ( V /. .~ ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   {csn 3694   <.cop 3697   U_ciun 3996   class class class wbr 4114   |-> cmpt 4176   X. cxp 4752   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   Er wer 6777   [cec 6778   /.cqs 6779   Basecbs 13296   /._s cqus 13566

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-er 6780  df-ec 6782  df-qs 6786  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302

This theorem is referenced by:  qusaddf  13600  qusmulf  13602