Theorem qusval 16202

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 16169	. . . 4 |- /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) )
3	2	a1i 11	. . 3 |- ( ph -> /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) ) )
4		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> r = R )
5	4	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( Base ` r ) = ( Base ` R ) )
6		qusval.v	. . . . . . . 8 |- ( ph -> V = ( Base ` R ) )
7	6	adantr 481	. . . . . . 7 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> V = ( Base ` R ) )
8	5, 7	eqtr4d 2659	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( Base ` r ) = V )
9		eceq2 7784	. . . . . . 7 |- ( e = .~ -> [ x ] e = [ x ] .~ )
10	9	ad2antll 765	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> [ x ] e = [ x ] .~ )
11	8, 10	mpteq12dv 4733	. . . . 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	syl6eqr 2674	. . . 4 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( x e. ( Base ` r ) |-> [ x ] e ) = F )
14	13, 4	oveq12d 6668	. . 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		elex 3212	. . . 4 |- ( R e. Z -> R e. _V )
17	15, 16	syl 17	. . 3 |- ( ph -> R e. _V )
18		qusval.e	. . . 4 |- ( ph -> .~ e. W )
19		elex 3212	. . . 4 |- ( .~ e. W -> .~ e. _V )
20	18, 19	syl 17	. . 3 |- ( ph -> .~ e. _V )
21		ovexd 6680	. . 3 |- ( ph -> ( F "s R ) e. _V )
22	3, 14, 17, 20, 21	ovmpt2d 6788	. 2 |- ( ph -> ( R /.s .~ ) = ( F "s R ) )
23	1, 22	eqtrd 2656	1 |- ( ph -> U = ( F "s R ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   [cec 7740   Basecbs 15857   "_s cimas 16164   /._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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ec 7744  df-qus 16169

This theorem is referenced by:  qusin  16204  qusbas  16205  quss  16206  qusaddval  16213  qusaddf  16214  qusmulval  16215  qusmulf  16216  qusgrp2  17533  qusring2  18620  qustps  21525  qustgpopn  21923  qustgplem  21924  qustgphaus  21926