Theorem qusval 13469

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 13449	. . . 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 531	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> r = R )
5	4	fveq2d 5652	. . . . . . 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 2267	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( Base ` r ) = V )
9		eceq2 6782	. . . . . . 7 |- ( e = .~ -> [ x ] e = [ x ] .~ )
10	9	ad2antll 491	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> [ x ] e = [ x ] .~ )
11	8, 10	mpteq12dv 4176	. . . . 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 2282	. . . 4 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( x e. ( Base ` r ) |-> [ x ] e ) = F )
14	13, 4	oveq12d 6046	. . 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 2817	. . 3 |- ( ph -> R e. _V )
17		qusval.e	. . . 4 |- ( ph -> .~ e. W )
18	17	elexd 2817	. . 3 |- ( ph -> .~ e. _V )
19		basfn 13204	. . . . . . . 8 |- Base Fn _V
20		funfvex 5665	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
21	20	funfni 5439	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
22	19, 16, 21	sylancr 414	. . . . . . 7 |- ( ph -> ( Base ` R ) e. _V )
23	6, 22	eqeltrd 2308	. . . . . 6 |- ( ph -> V e. _V )
24	23	mptexd 5891	. . . . 5 |- ( ph -> ( x e. V |-> [ x ] .~ ) e. _V )
25	12, 24	eqeltrid 2318	. . . 4 |- ( ph -> F e. _V )
26		imasex 13451	. . . 4 |- ( ( F e. _V /\ R e. Z ) -> ( F "s R ) e. _V )
27	25, 15, 26	syl2anc 411	. . 3 |- ( ph -> ( F "s R ) e. _V )
28	3, 14, 16, 18, 27	ovmpod 6159	. 2 |- ( ph -> ( R /.s .~ ) = ( F "s R ) )
29	1, 28	eqtrd 2264	1 |- ( ph -> U = ( F "s R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   |-> cmpt 4155   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   [cec 6743   Basecbs 13145   "_s cimas 13445   /._s cqus 13446

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-in1 619  ax-in2 620  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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-ec 6747  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-mulr 13237  df-iimas 13448  df-qus 13449

This theorem is referenced by:  qusin  13472  qusbas  13473  qusaddval  13481  qusaddf  13482  qusmulval  13483  qusmulf  13484  qusgrp2  13763  qusrng  14035  qusring2  14143