Theorem qusval 12762

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 12742	. . . 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 5531	. . . . . . 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 2223	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( Base ` r ) = V )
9		eceq2 6586	. . . . . . 7 |- ( e = .~ -> [ x ] e = [ x ] .~ )
10	9	ad2antll 491	. . . . . 6 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> [ x ] e = [ x ] .~ )
11	8, 10	mpteq12dv 4097	. . . . 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 2238	. . . 4 |- ( ( ph /\ ( r = R /\ e = .~ ) ) -> ( x e. ( Base ` r ) |-> [ x ] e ) = F )
14	13, 4	oveq12d 5906	. . 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 2762	. . 3 |- ( ph -> R e. _V )
17		qusval.e	. . . 4 |- ( ph -> .~ e. W )
18	17	elexd 2762	. . 3 |- ( ph -> .~ e. _V )
19		basfn 12534	. . . . . . . 8 |- Base Fn _V
20		funfvex 5544	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
21	20	funfni 5328	. . . . . . . 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 2264	. . . . . 6 |- ( ph -> V e. _V )
24	23	mptexd 5756	. . . . 5 |- ( ph -> ( x e. V |-> [ x ] .~ ) e. _V )
25	12, 24	eqeltrid 2274	. . . 4 |- ( ph -> F e. _V )
26		imasex 12744	. . . 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 6016	. 2 |- ( ph -> ( R /.s .~ ) = ( F "s R ) )
29	1, 28	eqtrd 2220	1 |- ( ph -> U = ( F "s R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   _Vcvv 2749   |-> cmpt 4076   Fn wfn 5223   ` cfv 5228  (class class class)co 5888   e. cmpo 5890   [cec 6547   Basecbs 12476   "_s cimas 12738   /._s cqus 12739

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1re 7919  ax-addrcl 7922

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-tp 3612  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-ec 6551  df-inn 8934  df-2 8992  df-3 8993  df-ndx 12479  df-slot 12480  df-base 12482  df-plusg 12564  df-mulr 12565  df-iimas 12741  df-qus 12742

This theorem is referenced by:  qusin  12765  qusbas  12766  qusaddval  12773  qusaddf  12774  qusmulval  12775  qusmulf  12776  qusgrp2  13008  qusrng  13210  qusring2  13314