Theorem qusval 13396

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 13376	. . . 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 529	. . . . . . . 8 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> r = R )
5	4	fveq2d 5639	. . . . . . 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 2265	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( Base ` r ) = V )
9		eceq2 6734	. . . . . . 7 |- ( e = .~ -> [ x ] e = [ x ] .~ )
10	9	ad2antll 491	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> [ x ] e = [ x ] .~ )
11	8, 10	mpteq12dv 4169	. . . . 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 2280	. . . 4 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( x e. ( Base ` r ) |-> [ x ] e ) = F )
14	13, 4	oveq12d 6031	. . 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 2814	. . 3 |- ( ph -> R e. _V )
17		qusval.e	. . . 4 |- ( ph -> .~ e. W )
18	17	elexd 2814	. . 3 |- ( ph -> .~ e. _V )
19		basfn 13131	. . . . . . . 8 |- Base Fn _V
20		funfvex 5652	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
21	20	funfni 5429	. . . . . . . 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 2306	. . . . . 6 |- ( ph -> V e. _V )
24	23	mptexd 5876	. . . . 5 |- ( ph -> ( x e. V |-> [ x ] .~ ) e. _V )
25	12, 24	eqeltrid 2316	. . . 4 |- ( ph -> F e. _V )
26		imasex 13378	. . . 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 6144	. 2 |- ( ph -> ( R /.s .~ ) = ( F "s R ) )
29	1, 28	eqtrd 2262	1 |- ( ph -> U = ( F "s R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   |-> cmpt 4148   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   e. cmpo 6015   [cec 6695   Basecbs 13072   "_s cimas 13372   /._s cqus 13373

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-tp 3675  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-ec 6699  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-mulr 13164  df-iimas 13375  df-qus 13376

This theorem is referenced by:  qusin  13399  qusbas  13400  qusaddval  13408  qusaddf  13409  qusmulval  13410  qusmulf  13411  qusgrp2  13690  qusrng  13961  qusring2  14069