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Theorem eceqoveq 6763
Description: Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
eceqoveq.5  |-  .~  Er  ( S  X.  S
)
eceqoveq.7  |-  dom  .+  =  ( S  X.  S )
eceqoveq.8  |-  -.  (/)  e.  S
eceqoveq.9  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
eceqoveq.10  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .~  <. C ,  D >.  <-> 
( A  .+  D
)  =  ( B 
.+  C ) ) )
Assertion
Ref Expression
eceqoveq  |-  ( ( A  e.  S  /\  C  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
Distinct variable groups:    x, y,  .+    x, S, y    x, A, y    x, B, y   
x, C, y    x, D, y
Allowed substitution hints:    .~ ( x, y)

Proof of Theorem eceqoveq
StepHypRef Expression
1 opelxpi 4721 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( S  X.  S
) )
21ad2antrr 706 . . . . . . 7  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  <. A ,  B >.  e.  ( S  X.  S ) )
3 eceqoveq.5 . . . . . . . . 9  |-  .~  Er  ( S  X.  S
)
43a1i 10 . . . . . . . 8  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  .~  Er  ( S  X.  S ) )
5 simpr 447 . . . . . . . 8  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )
64, 5ereldm 6703 . . . . . . 7  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  ( <. A ,  B >.  e.  ( S  X.  S )  <->  <. C ,  D >.  e.  ( S  X.  S ) ) )
72, 6mpbid 201 . . . . . 6  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  <. C ,  D >.  e.  ( S  X.  S ) )
8 opelxp2 4723 . . . . . 6  |-  ( <. C ,  D >.  e.  ( S  X.  S
)  ->  D  e.  S )
97, 8syl 15 . . . . 5  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  D  e.  S
)
109ex 423 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  D  e.  S ) )
11 eceqoveq.9 . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
1211caovcl 6014 . . . . . . 7  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( B  .+  C
)  e.  S )
13 eleq1 2343 . . . . . . 7  |-  ( ( A  .+  D )  =  ( B  .+  C )  ->  (
( A  .+  D
)  e.  S  <->  ( B  .+  C )  e.  S
) )
1412, 13syl5ibr 212 . . . . . 6  |-  ( ( A  .+  D )  =  ( B  .+  C )  ->  (
( B  e.  S  /\  C  e.  S
)  ->  ( A  .+  D )  e.  S
) )
15 eceqoveq.7 . . . . . . . 8  |-  dom  .+  =  ( S  X.  S )
16 eceqoveq.8 . . . . . . . 8  |-  -.  (/)  e.  S
1715, 16ndmovrcl 6006 . . . . . . 7  |-  ( ( A  .+  D )  e.  S  ->  ( A  e.  S  /\  D  e.  S )
)
1817simprd 449 . . . . . 6  |-  ( ( A  .+  D )  e.  S  ->  D  e.  S )
1914, 18syl6com 31 . . . . 5  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( ( A  .+  D )  =  ( B  .+  C )  ->  D  e.  S
) )
2019adantll 694 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S )  ->  (
( A  .+  D
)  =  ( B 
.+  C )  ->  D  e.  S )
)
213a1i 10 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  .~  Er  ( S  X.  S ) )
221adantr 451 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <. A ,  B >.  e.  ( S  X.  S
) )
2321, 22erth 6704 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .~  <. C ,  D >.  <->  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  ) )
24 eceqoveq.10 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .~  <. C ,  D >.  <-> 
( A  .+  D
)  =  ( B 
.+  C ) ) )
2523, 24bitr3d 246 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
2625expr 598 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S )  ->  ( D  e.  S  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) ) )
2710, 20, 26pm5.21ndd 343 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
2827an32s 779 . 2  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  B  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
29 eqcom 2285 . . . 4  |-  ( (/)  =  [ <. C ,  D >. ]  .~  <->  [ <. C ,  D >. ]  .~  =  (/) )
30 erdm 6670 . . . . . . . . . . . 12  |-  (  .~  Er  ( S  X.  S
)  ->  dom  .~  =  ( S  X.  S
) )
313, 30ax-mp 8 . . . . . . . . . . 11  |-  dom  .~  =  ( S  X.  S )
3231eleq2i 2347 . . . . . . . . . 10  |-  ( <. C ,  D >.  e. 
dom  .~  <->  <. C ,  D >.  e.  ( S  X.  S ) )
33 ecdmn0 6702 . . . . . . . . . 10  |-  ( <. C ,  D >.  e. 
dom  .~  <->  [ <. C ,  D >. ]  .~  =/=  (/) )
34 opelxp 4719 . . . . . . . . . 10  |-  ( <. C ,  D >.  e.  ( S  X.  S
)  <->  ( C  e.  S  /\  D  e.  S ) )
3532, 33, 343bitr3i 266 . . . . . . . . 9  |-  ( [
<. C ,  D >. ]  .~  =/=  (/)  <->  ( C  e.  S  /\  D  e.  S ) )
3635simplbi2 608 . . . . . . . 8  |-  ( C  e.  S  ->  ( D  e.  S  ->  [
<. C ,  D >. ]  .~  =/=  (/) ) )
3736ad2antlr 707 . . . . . . 7  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( D  e.  S  ->  [
<. C ,  D >. ]  .~  =/=  (/) ) )
3837necon2bd 2495 . . . . . 6  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. C ,  D >. ]  .~  =  (/)  ->  -.  D  e.  S
) )
39 simpr 447 . . . . . . . 8  |-  ( ( A  e.  S  /\  D  e.  S )  ->  D  e.  S )
4039con3i 127 . . . . . . 7  |-  ( -.  D  e.  S  ->  -.  ( A  e.  S  /\  D  e.  S
) )
4115ndmov 6004 . . . . . . 7  |-  ( -.  ( A  e.  S  /\  D  e.  S
)  ->  ( A  .+  D )  =  (/) )
4240, 41syl 15 . . . . . 6  |-  ( -.  D  e.  S  -> 
( A  .+  D
)  =  (/) )
4338, 42syl6 29 . . . . 5  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. C ,  D >. ]  .~  =  (/)  ->  ( A  .+  D
)  =  (/) ) )
44 eleq1 2343 . . . . . . 7  |-  ( ( A  .+  D )  =  (/)  ->  ( ( A  .+  D )  e.  S  <->  (/)  e.  S
) )
4516, 44mtbiri 294 . . . . . 6  |-  ( ( A  .+  D )  =  (/)  ->  -.  ( A  .+  D )  e.  S )
4635simprbi 450 . . . . . . . 8  |-  ( [
<. C ,  D >. ]  .~  =/=  (/)  ->  D  e.  S )
4711caovcl 6014 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  D  e.  S )  ->  ( A  .+  D
)  e.  S )
4847ex 423 . . . . . . . . 9  |-  ( A  e.  S  ->  ( D  e.  S  ->  ( A  .+  D )  e.  S ) )
4948ad2antrr 706 . . . . . . . 8  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( D  e.  S  ->  ( A  .+  D )  e.  S ) )
5046, 49syl5 28 . . . . . . 7  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. C ,  D >. ]  .~  =/=  (/)  ->  ( A  .+  D )  e.  S ) )
5150necon1bd 2514 . . . . . 6  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( -.  ( A  .+  D
)  e.  S  ->  [ <. C ,  D >. ]  .~  =  (/) ) )
5245, 51syl5 28 . . . . 5  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  (
( A  .+  D
)  =  (/)  ->  [ <. C ,  D >. ]  .~  =  (/) ) )
5343, 52impbid 183 . . . 4  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. C ,  D >. ]  .~  =  (/)  <->  ( A  .+  D )  =  (/) ) )
5429, 53syl5bb 248 . . 3  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( (/)  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  (/) ) )
5531eleq2i 2347 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  .~  <->  <. A ,  B >.  e.  ( S  X.  S ) )
56 ecdmn0 6702 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  .~  <->  [ <. A ,  B >. ]  .~  =/=  (/) )
57 opelxp 4719 . . . . . . . 8  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
5855, 56, 573bitr3i 266 . . . . . . 7  |-  ( [
<. A ,  B >. ]  .~  =/=  (/)  <->  ( A  e.  S  /\  B  e.  S ) )
5958simprbi 450 . . . . . 6  |-  ( [
<. A ,  B >. ]  .~  =/=  (/)  ->  B  e.  S )
6059necon1bi 2489 . . . . 5  |-  ( -.  B  e.  S  ->  [ <. A ,  B >. ]  .~  =  (/) )
6160adantl 452 . . . 4  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  [ <. A ,  B >. ]  .~  =  (/) )
6261eqeq1d 2291 . . 3  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  (/)  =  [ <. C ,  D >. ]  .~  ) )
63 simpl 443 . . . . . . 7  |-  ( ( B  e.  S  /\  C  e.  S )  ->  B  e.  S )
6463con3i 127 . . . . . 6  |-  ( -.  B  e.  S  ->  -.  ( B  e.  S  /\  C  e.  S
) )
6515ndmov 6004 . . . . . 6  |-  ( -.  ( B  e.  S  /\  C  e.  S
)  ->  ( B  .+  C )  =  (/) )
6664, 65syl 15 . . . . 5  |-  ( -.  B  e.  S  -> 
( B  .+  C
)  =  (/) )
6766adantl 452 . . . 4  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( B  .+  C )  =  (/) )
6867eqeq2d 2294 . . 3  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  (
( A  .+  D
)  =  ( B 
.+  C )  <->  ( A  .+  D )  =  (/) ) )
6954, 62, 683bitr4d 276 . 2  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
7028, 69pm2.61dan 766 1  |-  ( ( A  e.  S  /\  C  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   <.cop 3643   class class class wbr 4023    X. cxp 4687   dom cdm 4689  (class class class)co 5858    Er wer 6657   [cec 6658
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-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-fv 5263  df-ov 5861  df-er 6660  df-ec 6662
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