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Theorem divsval 14485
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 14452 . . . 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 755 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
r  =  R )
54fveq2d 5700 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  ( Base `  R
) )
6 divsval.v . . . . . . . 8  |-  ( ph  ->  V  =  ( Base `  R ) )
76adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  V  =  ( Base `  R ) )
85, 7eqtr4d 2478 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  V )
9 eceq2 7143 . . . . . . 7  |-  ( e  =  .~  ->  [ x ] e  =  [
x ]  .~  )
109ad2antll 728 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  [ x ] e  =  [ x ]  .~  )
118, 10mpteq12dv 4375 . . . . 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 2493 . . . 4  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( x  e.  (
Base `  r )  |->  [ x ] e )  =  F )
1413, 4oveq12d 6114 . . 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 2986 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
1715, 16syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
18 divsval.e . . . 4  |-  ( ph  ->  .~  e.  W )
19 elex 2986 . . . 4  |-  (  .~  e.  W  ->  .~  e.  _V )
2018, 19syl 16 . . 3  |-  ( ph  ->  .~  e.  _V )
21 ovex 6121 . . . 4  |-  ( F 
"s  R )  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( F  "s  R )  e.  _V )
233, 14, 17, 20, 22ovmpt2d 6223 . 2  |-  ( ph  ->  ( R  /.s  .~  )  =  ( F  "s  R
) )
241, 23eqtrd 2475 1  |-  ( ph  ->  U  =  ( F 
"s  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977    e. cmpt 4355   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   [cec 7104   Basecbs 14179    "s cimas 14447    /.s cqus 14448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-ec 7108  df-divs 14452
This theorem is referenced by:  divsin  14487  divsbas  14488  divssca  14489  divsaddval  14496  divsaddf  14497  divsmulval  14498  divsmulf  14499  divsgrp2  15678  divsrng2  16717  divstps  19300  divstgpopn  19695  divstgplem  19696  divstgphaus  19698
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