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Theorem divsval 13444
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 13412 . . . 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 5529 . . . . . . 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 2318 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  V )
9 eceq2 6697 . . . . . . 7  |-  ( e  =  .~  ->  [ x ] e  =  [
x ]  .~  )
109ad2antll 709 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  [ x ] e  =  [ x ]  .~  )
118, 10mpteq12dv 4098 . . . . 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 2333 . . . 4  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( x  e.  (
Base `  r )  |->  [ x ] e )  =  F )
1413, 4oveq12d 5876 . . 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 2796 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
1715, 16syl 15 . . 3  |-  ( ph  ->  R  e.  _V )
18 divsval.e . . . 4  |-  ( ph  ->  .~  e.  W )
19 elex 2796 . . . 4  |-  (  .~  e.  W  ->  .~  e.  _V )
2018, 19syl 15 . . 3  |-  ( ph  ->  .~  e.  _V )
21 ovex 5883 . . . 4  |-  ( F 
"s  R )  e.  _V
2221a1i 10 . . 3  |-  ( ph  ->  ( F  "s  R )  e.  _V )
233, 14, 17, 20, 22ovmpt2d 5975 . 2  |-  ( ph  ->  ( R  /.s  .~  )  =  ( F  "s  R
) )
241, 23eqtrd 2315 1  |-  ( ph  ->  U  =  ( F 
"s  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   [cec 6658   Basecbs 13148    "s cimas 13407    /.s cqus 13408
This theorem is referenced by:  divsin  13446  divsbas  13447  divssca  13448  divsaddval  13455  divsaddf  13456  divsmulval  13457  divsmulf  13458  divsgrp2  14613  divsrng2  15403  divstps  17413  divstgpopn  17802  divstgplem  17803  divstgphaus  17805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ec 6662  df-divs 13412
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