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Theorem divsval 13460
Description: Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
divsval.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
divsval.v  |-  ( ph  ->  V  =  ( Base `  R ) )
divsval.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
divsval.e  |-  ( ph  ->  .~  e.  W )
divsval.r  |-  ( ph  ->  R  e.  Z )
Assertion
Ref Expression
divsval  |-  ( 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 divsval
Dummy variables  e 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 divsval.u . 2  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
2 df-divs 13428 . . . 4  |-  /.s  =  (
r  e.  _V , 
e  e.  _V  |->  ( ( x  e.  (
Base `  r )  |->  [ x ] e )  "s  r ) )
32a1i 10 . . 3  |-  ( ph  ->  /.s  =  ( r  e. 
_V ,  e  e. 
_V  |->  ( ( x  e.  ( Base `  r
)  |->  [ x ]
e )  "s  r )
) )
4 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
r  =  R )
54fveq2d 5545 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  ( Base `  R
) )
6 divsval.v . . . . . . . 8  |-  ( ph  ->  V  =  ( Base `  R ) )
76adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  V  =  ( Base `  R ) )
85, 7eqtr4d 2331 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  V )
9 eceq2 6713 . . . . . . 7  |-  ( e  =  .~  ->  [ x ] e  =  [
x ]  .~  )
109ad2antll 709 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  [ x ] e  =  [ x ]  .~  )
118, 10mpteq12dv 4114 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( x  e.  (
Base `  r )  |->  [ x ] e )  =  ( x  e.  V  |->  [ x ]  .~  ) )
12 divsval.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
1311, 12syl6eqr 2346 . . . 4  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( x  e.  (
Base `  r )  |->  [ x ] e )  =  F )
1413, 4oveq12d 5892 . . 3  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( ( x  e.  ( Base `  r
)  |->  [ x ]
e )  "s  r )  =  ( F  "s  R
) )
15 divsval.r . . . 4  |-  ( ph  ->  R  e.  Z )
16 elex 2809 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
1715, 16syl 15 . . 3  |-  ( ph  ->  R  e.  _V )
18 divsval.e . . . 4  |-  ( ph  ->  .~  e.  W )
19 elex 2809 . . . 4  |-  (  .~  e.  W  ->  .~  e.  _V )
2018, 19syl 15 . . 3  |-  ( ph  ->  .~  e.  _V )
21 ovex 5899 . . . 4  |-  ( F 
"s  R )  e.  _V
2221a1i 10 . . 3  |-  ( ph  ->  ( F  "s  R )  e.  _V )
233, 14, 17, 20, 22ovmpt2d 5991 . 2  |-  ( ph  ->  ( R  /.s  .~  )  =  ( F  "s  R
) )
241, 23eqtrd 2328 1  |-  ( ph  ->  U  =  ( F 
"s  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   [cec 6674   Basecbs 13164    "s cimas 13423    /.s cqus 13424
This theorem is referenced by:  divsin  13462  divsbas  13463  divssca  13464  divsaddval  13471  divsaddf  13472  divsmulval  13473  divsmulf  13474  divsgrp2  14629  divsrng2  15419  divstps  17429  divstgpopn  17818  divstgplem  17819  divstgphaus  17821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ec 6678  df-divs 13428
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