MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divsin Unicode version

Theorem divsin 13462
Description: Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
divsin.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
divsin.v  |-  ( ph  ->  V  =  ( Base `  R ) )
divsin.e  |-  ( ph  ->  .~  e.  W )
divsin.r  |-  ( ph  ->  R  e.  Z )
divsin.s  |-  ( ph  ->  (  .~  " V
)  C_  V )
Assertion
Ref Expression
divsin  |-  ( ph  ->  U  =  ( R 
/.s  (  .~  i^i  ( V  X.  V ) ) ) )

Proof of Theorem divsin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divsin.s . . . . 5  |-  ( ph  ->  (  .~  " V
)  C_  V )
2 ecinxp 6750 . . . . 5  |-  ( ( (  .~  " V
)  C_  V  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V ) ) )
31, 2sylan 457 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V
) ) )
43mpteq2dva 4122 . . 3  |-  ( ph  ->  ( x  e.  V  |->  [ x ]  .~  )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) ) )
54oveq1d 5889 . 2  |-  ( ph  ->  ( ( x  e.  V  |->  [ x ]  .~  )  "s  R )  =  ( ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V ) ) )  "s  R ) )
6 divsin.u . . 3  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
7 divsin.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
8 eqid 2296 . . 3  |-  ( x  e.  V  |->  [ x ]  .~  )  =  ( x  e.  V  |->  [ x ]  .~  )
9 divsin.e . . 3  |-  ( ph  ->  .~  e.  W )
10 divsin.r . . 3  |-  ( ph  ->  R  e.  Z )
116, 7, 8, 9, 10divsval 13460 . 2  |-  ( ph  ->  U  =  ( ( x  e.  V  |->  [ x ]  .~  )  "s  R ) )
12 eqidd 2297 . . 3  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) ) )
13 eqid 2296 . . 3  |-  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V ) ) )
14 inex1g 4173 . . . 4  |-  (  .~  e.  W  ->  (  .~  i^i  ( V  X.  V
) )  e.  _V )
159, 14syl 15 . . 3  |-  ( ph  ->  (  .~  i^i  ( V  X.  V ) )  e.  _V )
1612, 7, 13, 15, 10divsval 13460 . 2  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( ( x  e.  V  |->  [ x ]
(  .~  i^i  ( V  X.  V ) ) )  "s  R ) )
175, 11, 163eqtr4d 2338 1  |-  ( ph  ->  U  =  ( R 
/.s  (  .~  i^i  ( V  X.  V ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165    e. cmpt 4093    X. cxp 4703   "cima 4708   ` cfv 5271  (class class class)co 5874   [cec 6674   Basecbs 13164    "s cimas 13423    /.s cqus 13424
This theorem is referenced by:  pi1addf  18561  pi1addval  18562  pi1grplem  18563
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
  Copyright terms: Public domain W3C validator