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Theorem ercpbl 13451
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 976 . 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 976 . . . 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 976 . . . 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 981 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  A  e.  V )
94, 6, 7, 8ercpbllem 13450 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  A )  =  ( F `  C )  <->  A  .~  C ) )
10 simp2r 982 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  B  e.  V )
114, 6, 7, 10ercpbllem 13450 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( ( F `  B )  =  ( F `  D )  <->  B  .~  D ) )
129, 11anbi12d 691 . 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 6012 . . . 4  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A  .+  B
)  e.  V )
15143adant3 975 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( A  .+  B )  e.  V
)
164, 6, 7, 15ercpbllem 13450 . 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 259 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    Er wer 6657   [cec 6658
This theorem is referenced by:  divsaddvallem  13453  divsaddflem  13454  divsgrp2  14613  divsrng2  15403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-er 6660  df-ec 6662
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