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Theorem divsval 13694
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 13662 . . . 4  |-  /.s  =  (
r  e.  _V , 
e  e.  _V  |->  ( ( x  e.  (
Base `  r )  |->  [ x ] e )  "s  r ) )
32a1i 11 . . 3  |-  ( ph  ->  /.s  =  ( r  e. 
_V ,  e  e. 
_V  |->  ( ( x  e.  ( Base `  r
)  |->  [ x ]
e )  "s  r )
) )
4 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
r  =  R )
54fveq2d 5672 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  ( Base `  R
) )
6 divsval.v . . . . . . . 8  |-  ( ph  ->  V  =  ( Base `  R ) )
76adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  V  =  ( Base `  R ) )
85, 7eqtr4d 2422 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  V )
9 eceq2 6878 . . . . . . 7  |-  ( e  =  .~  ->  [ x ] e  =  [
x ]  .~  )
109ad2antll 710 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  [ x ] e  =  [ x ]  .~  )
118, 10mpteq12dv 4228 . . . . 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 2437 . . . 4  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( x  e.  (
Base `  r )  |->  [ x ] e )  =  F )
1413, 4oveq12d 6038 . . 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 2907 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
1715, 16syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
18 divsval.e . . . 4  |-  ( ph  ->  .~  e.  W )
19 elex 2907 . . . 4  |-  (  .~  e.  W  ->  .~  e.  _V )
2018, 19syl 16 . . 3  |-  ( ph  ->  .~  e.  _V )
21 ovex 6045 . . . 4  |-  ( F 
"s  R )  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( F  "s  R )  e.  _V )
233, 14, 17, 20, 22ovmpt2d 6140 . 2  |-  ( ph  ->  ( R  /.s  .~  )  =  ( F  "s  R
) )
241, 23eqtrd 2419 1  |-  ( ph  ->  U  =  ( F 
"s  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    e. cmpt 4207   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   [cec 6839   Basecbs 13396    "s cimas 13657    /.s cqus 13658
This theorem is referenced by:  divsin  13696  divsbas  13697  divssca  13698  divsaddval  13705  divsaddf  13706  divsmulval  13707  divsmulf  13708  divsgrp2  14863  divsrng2  15653  divstps  17675  divstgpopn  18070  divstgplem  18071  divstgphaus  18073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-ec 6843  df-divs 13662
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