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Theorem qusval 12766
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.
StepHypRef Expression
1 qusval.u . 2  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
2 df-qus 12746 . . . 4  |-  /.s  =  (
r  e.  _V , 
e  e.  _V  |->  ( ( x  e.  (
Base `  r )  |->  [ x ] e )  "s  r ) )
32a1i 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 )
54fveq2d 5534 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  ( Base `  R
) )
6 qusval.v . . . . . . . 8  |-  ( ph  ->  V  =  ( Base `  R ) )
76adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  V  =  ( Base `  R ) )
85, 7eqtr4d 2225 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( Base `  r )  =  V )
9 eceq2 6590 . . . . . . 7  |-  ( e  =  .~  ->  [ x ] e  =  [
x ]  .~  )
109ad2antll 491 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  ->  [ x ] e  =  [ x ]  .~  )
118, 10mpteq12dv 4100 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( x  e.  (
Base `  r )  |->  [ x ] e )  =  ( x  e.  V  |->  [ x ]  .~  ) )
12 qusval.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
1311, 12eqtr4di 2240 . . . 4  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( x  e.  (
Base `  r )  |->  [ x ] e )  =  F )
1413, 4oveq12d 5909 . . 3  |-  ( (
ph  /\  ( r  =  R  /\  e  =  .~  ) )  -> 
( ( x  e.  ( Base `  r
)  |->  [ x ]
e )  "s  r )  =  ( F  "s  R
) )
15 qusval.r . . . 4  |-  ( ph  ->  R  e.  Z )
1615elexd 2765 . . 3  |-  ( ph  ->  R  e.  _V )
17 qusval.e . . . 4  |-  ( ph  ->  .~  e.  W )
1817elexd 2765 . . 3  |-  ( ph  ->  .~  e.  _V )
19 basfn 12538 . . . . . . . 8  |-  Base  Fn  _V
20 funfvex 5547 . . . . . . . . 9  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
2120funfni 5331 . . . . . . . 8  |-  ( (
Base  Fn  _V  /\  R  e.  _V )  ->  ( Base `  R )  e. 
_V )
2219, 16, 21sylancr 414 . . . . . . 7  |-  ( ph  ->  ( Base `  R
)  e.  _V )
236, 22eqeltrd 2266 . . . . . 6  |-  ( ph  ->  V  e.  _V )
2423mptexd 5759 . . . . 5  |-  ( ph  ->  ( x  e.  V  |->  [ x ]  .~  )  e.  _V )
2512, 24eqeltrid 2276 . . . 4  |-  ( ph  ->  F  e.  _V )
26 imasex 12748 . . . 4  |-  ( ( F  e.  _V  /\  R  e.  Z )  ->  ( F  "s  R )  e.  _V )
2725, 15, 26syl2anc 411 . . 3  |-  ( ph  ->  ( F  "s  R )  e.  _V )
283, 14, 16, 18, 27ovmpod 6019 . 2  |-  ( ph  ->  ( R  /.s  .~  )  =  ( F  "s  R
) )
291, 28eqtrd 2222 1  |-  ( ph  ->  U  =  ( F 
"s  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752    |-> cmpt 4079    Fn wfn 5226   ` cfv 5231  (class class class)co 5891    e. cmpo 5893   [cec 6551   Basecbs 12480    "s cimas 12742    /.s cqus 12743
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-tp 3615  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-ec 6555  df-inn 8938  df-2 8996  df-3 8997  df-ndx 12483  df-slot 12484  df-base 12486  df-plusg 12568  df-mulr 12569  df-iimas 12745  df-qus 12746
This theorem is referenced by:  qusin  12769  qusbas  12770  qusaddval  12777  qusaddf  12778  qusmulval  12779  qusmulf  12780  qusgrp2  13021  qusrng  13273  qusring2  13377
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