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Theorem ercpbl 14965
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 1017 . 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 1017 . . . 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 1017 . . . 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 1022 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  A  e.  V )
94, 6, 7, 8ercpbllem 14964 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  A )  =  ( F `  C )  <->  A  .~  C ) )
10 simp2r 1023 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  B  e.  V )
114, 6, 7, 10ercpbllem 14964 . . 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 6466 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A  .+  B
)  e.  V )
15143adant3 1016 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( A  .+  B )  e.  V
)
164, 6, 7, 15ercpbllem 14964 . 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    Er wer 7326   [cec 7327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-er 7329  df-ec 7331
This theorem is referenced by:  qusaddvallem  14967  qusaddflem  14968  qusgrp2  16314  qusring2  17395
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