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

Theorem ercpbl 13701
Description: Translate the function compatiblity relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ercpbl.r  |-  ( ph  ->  .~  Er  V )
ercpbl.v  |-  ( ph  ->  V  e.  _V )
ercpbl.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
ercpbl.c  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V ) )  -> 
( a  .+  b
)  e.  V )
ercpbl.e  |-  ( ph  ->  ( ( A  .~  C  /\  B  .~  D
)  ->  ( A  .+  B )  .~  ( C  .+  D ) ) )
Assertion
Ref Expression
ercpbl  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  -> 
( F `  ( A  .+  B ) )  =  ( F `  ( C  .+  D ) ) ) )
Distinct variable groups:    x,  .~    a, b, x, A    B, b, x    x, C    x, D    V, a, b, x    .+ , a, b, x    ph, a,
b, x
Allowed substitution hints:    B( a)    C( a, b)    D( a, b)    .~ ( a, b)    F( x, a, b)

Proof of Theorem ercpbl
StepHypRef Expression
1 ercpbl.e . . 3  |-  ( ph  ->  ( ( A  .~  C  /\  B  .~  D
)  ->  ( A  .+  B )  .~  ( C  .+  D ) ) )
213ad2ant1 978 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( A  .~  C  /\  B  .~  D )  ->  ( A  .+  B )  .~  ( C  .+  D ) ) )
3 ercpbl.r . . . . 5  |-  ( ph  ->  .~  Er  V )
433ad2ant1 978 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  .~  Er  V
)
5 ercpbl.v . . . . 5  |-  ( ph  ->  V  e.  _V )
653ad2ant1 978 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  V  e.  _V )
7 ercpbl.f . . . 4  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
8 simp2l 983 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  A  e.  V )
94, 6, 7, 8ercpbllem 13700 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  A )  =  ( F `  C )  <->  A  .~  C ) )
10 simp2r 984 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  B  e.  V )
114, 6, 7, 10ercpbllem 13700 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  B )  =  ( F `  D )  <->  B  .~  D ) )
129, 11anbi12d 692 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  <->  ( A  .~  C  /\  B  .~  D ) ) )
13 ercpbl.c . . . . 5  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V ) )  -> 
( a  .+  b
)  e.  V )
1413caovclg 6178 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A  .+  B
)  e.  V )
15143adant3 977 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( A  .+  B )  e.  V
)
164, 6, 7, 15ercpbllem 13700 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  ( A  .+  B ) )  =  ( F `  ( C  .+  D ) )  <-> 
( A  .+  B
)  .~  ( C  .+  D ) ) )
172, 12, 163imtr4d 260 1  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  -> 
( F `  ( A  .+  B ) )  =  ( F `  ( C  .+  D ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2899   class class class wbr 4153    e. cmpt 4207   ` cfv 5394  (class class class)co 6020    Er wer 6838   [cec 6839
This theorem is referenced by:  divsaddvallem  13703  divsaddflem  13704  divsgrp2  14863  divsrng2  15653
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-13 1719  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-pow 4318  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-er 6841  df-ec 6843
  Copyright terms: Public domain W3C validator