Theorem qusval 13703

Description: Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusval.u	|- ( ph -> U = ( R /.s .~ ) )
qusval.v	|- ( ph -> V = ( Base ` R ) )
qusval.f	|- F = ( x e. V |-> [ x ] .~ )
qusval.e	|- ( ph -> .~ e. W )
qusval.r	|- ( ph -> R e. Z )

Assertion

Ref	Expression
qusval	|- ( ph -> U = ( F "s R ) )

Distinct variable groups:   x, .~   ph, x   x, R   x, V

Allowed substitution hints:   U( x)   F( x)   W( x)   Z( x)

Proof of Theorem qusval

Dummy variables e r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusval.u	. 2 |- ( ph -> U = ( R /.s .~ ) )
2		df-qus 13684	. . . 4 |- /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) )
3	2	a1i 9	. . 3 |- ( ph -> /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) ) )
4		simprl 535	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> r = R )
5	4	fveq2d 5702	. . . . . . 7 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( Base ` r ) = ( Base ` R ) )
6		qusval.v	. . . . . . . 8 |- ( ph -> V = ( Base ` R ) )
7	6	adantr 276	. . . . . . 7 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> V = ( Base ` R ) )
8	5, 7	eqtr4d 2274	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( Base ` r ) = V )
9		eceq2 6848	. . . . . . 7 |- ( e = .~ -> [ x ] e = [ x ] .~ )
10	9	ad2antll 495	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> [ x ] e = [ x ] .~ )
11	8, 10	mpteq12dv 4214	. . . . 5 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( x e. ( Base ` r ) |-> [ x ] e ) = ( x e. V |-> [ x ] .~ ) )
12		qusval.f	. . . . 5 |- F = ( x e. V |-> [ x ] .~ )
13	11, 12	eqtr4di 2289	. . . 4 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( x e. ( Base ` r ) |-> [ x ] e ) = F )
14	13, 4	oveq12d 6106	. . 3 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) = ( F "s R ) )
15		qusval.r	. . . 4 |- ( ph -> R e. Z )
16	15	elexd 2835	. . 3 |- ( ph -> R e. _V )
17		qusval.e	. . . 4 |- ( ph -> .~ e. W )
18	17	elexd 2835	. . 3 |- ( ph -> .~ e. _V )
19		basfn 13469	. . . . . . . 8 |- Base Fn _V
20		funfvex 5715	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
21	20	funfni 5486	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
22	19, 16, 21	sylancr 418	. . . . . . 7 |- ( ph -> ( Base ` R ) e. _V )
23	6, 22	eqeltrd 2315	. . . . . 6 |- ( ph -> V e. _V )
24	23	mptexd 5947	. . . . 5 |- ( ph -> ( x e. V |-> [ x ] .~ ) e. _V )
25	12, 24	eqeltrid 2325	. . . 4 |- ( ph -> F e. _V )
26		imasex 13685	. . . 4 |- ( ( F e. _V /\ R e. Z ) -> ( F "s R ) e. _V )
27	25, 15, 26	syl2anc 415	. . 3 |- ( ph -> ( F "s R ) e. _V )
28	3, 14, 16, 18, 27	ovmpod 6219	. 2 |- ( ph -> ( R /.s .~ ) = ( F "s R ) )
29	1, 28	eqtrd 2271	1 |- ( ph -> U = ( F "s R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   |-> cmpt 4193   Fn wfn 5375   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   [cec 6809   Basecbs 13410   "_s cimas 13681   /._s cqus 13682

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-ec 6813  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-mulr 13504  df-iimas 13683  df-qus 13684

This theorem is used by:  qusin  13706  qusbas  13707  qusaddval  13715  qusaddf  13716  qusmulval  13717  qusmulf  13718  qusgrp2  13975  qusrng  14364  qusring2  14480