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

Theorem ercpbl 14492
Description: Translate the function compatibility 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 1009 . 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 1009 . . . 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 1009 . . . 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 1014 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  A  e.  V )
94, 6, 7, 8ercpbllem 14491 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  A )  =  ( F `  C )  <->  A  .~  C ) )
10 simp2r 1015 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  B  e.  V )
114, 6, 7, 10ercpbllem 14491 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  B )  =  ( F `  D )  <->  B  .~  D ) )
129, 11anbi12d 710 . 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 6260 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A  .+  B
)  e.  V )
15143adant3 1008 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( A  .+  B )  e.  V
)
164, 6, 7, 15ercpbllem 14491 . 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 268 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2977   class class class wbr 4297    e. cmpt 4355   ` cfv 5423  (class class class)co 6096    Er wer 7103   [cec 7104
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-8 1758  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-pow 4475  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-er 7106  df-ec 7108
This theorem is referenced by:  divsaddvallem  14494  divsaddflem  14495  divsgrp2  15678  divsrng2  16717
  Copyright terms: Public domain W3C validator