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Theorem eceqoveq 7001
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 4902 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( S  X.  S
) )
21ad2antrr 707 . . . . . . 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 11 . . . . . . . 8  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  .~  Er  ( S  X.  S ) )
5 simpr 448 . . . . . . . 8  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )
64, 5ereldm 6940 . . . . . . 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 202 . . . . . 6  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  <. C ,  D >.  e.  ( S  X.  S ) )
8 opelxp2 4904 . . . . . 6  |-  ( <. C ,  D >.  e.  ( S  X.  S
)  ->  D  e.  S )
97, 8syl 16 . . . . 5  |-  ( ( ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S )  /\  [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  D  e.  S
)
109ex 424 . . . 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 6233 . . . . . . 7  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( B  .+  C
)  e.  S )
13 eleq1 2495 . . . . . . 7  |-  ( ( A  .+  D )  =  ( B  .+  C )  ->  (
( A  .+  D
)  e.  S  <->  ( B  .+  C )  e.  S
) )
1412, 13syl5ibr 213 . . . . . 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 6225 . . . . . . 7  |-  ( ( A  .+  D )  e.  S  ->  ( A  e.  S  /\  D  e.  S )
)
1817simprd 450 . . . . . 6  |-  ( ( A  .+  D )  e.  S  ->  D  e.  S )
1914, 18syl6com 33 . . . . 5  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( ( A  .+  D )  =  ( B  .+  C )  ->  D  e.  S
) )
2019adantll 695 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S )  ->  (
( A  .+  D
)  =  ( B 
.+  C )  ->  D  e.  S )
)
213a1i 11 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  .~  Er  ( S  X.  S ) )
221adantr 452 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <. A ,  B >.  e.  ( S  X.  S
) )
2321, 22erth 6941 . . . . . 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 247 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
2625expr 599 . . . 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 344 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
2827an32s 780 . 2  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  B  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
29 eqcom 2437 . . . 4  |-  ( (/)  =  [ <. C ,  D >. ]  .~  <->  [ <. C ,  D >. ]  .~  =  (/) )
30 erdm 6907 . . . . . . . . . . . 12  |-  (  .~  Er  ( S  X.  S
)  ->  dom  .~  =  ( S  X.  S
) )
313, 30ax-mp 8 . . . . . . . . . . 11  |-  dom  .~  =  ( S  X.  S )
3231eleq2i 2499 . . . . . . . . . 10  |-  ( <. C ,  D >.  e. 
dom  .~  <->  <. C ,  D >.  e.  ( S  X.  S ) )
33 ecdmn0 6939 . . . . . . . . . 10  |-  ( <. C ,  D >.  e. 
dom  .~  <->  [ <. C ,  D >. ]  .~  =/=  (/) )
34 opelxp 4900 . . . . . . . . . 10  |-  ( <. C ,  D >.  e.  ( S  X.  S
)  <->  ( C  e.  S  /\  D  e.  S ) )
3532, 33, 343bitr3i 267 . . . . . . . . 9  |-  ( [
<. C ,  D >. ]  .~  =/=  (/)  <->  ( C  e.  S  /\  D  e.  S ) )
3635simplbi2 609 . . . . . . . 8  |-  ( C  e.  S  ->  ( D  e.  S  ->  [
<. C ,  D >. ]  .~  =/=  (/) ) )
3736ad2antlr 708 . . . . . . 7  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( D  e.  S  ->  [
<. C ,  D >. ]  .~  =/=  (/) ) )
3837necon2bd 2647 . . . . . 6  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. C ,  D >. ]  .~  =  (/)  ->  -.  D  e.  S
) )
39 simpr 448 . . . . . . . 8  |-  ( ( A  e.  S  /\  D  e.  S )  ->  D  e.  S )
4039con3i 129 . . . . . . 7  |-  ( -.  D  e.  S  ->  -.  ( A  e.  S  /\  D  e.  S
) )
4115ndmov 6223 . . . . . . 7  |-  ( -.  ( A  e.  S  /\  D  e.  S
)  ->  ( A  .+  D )  =  (/) )
4240, 41syl 16 . . . . . 6  |-  ( -.  D  e.  S  -> 
( A  .+  D
)  =  (/) )
4338, 42syl6 31 . . . . 5  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. C ,  D >. ]  .~  =  (/)  ->  ( A  .+  D
)  =  (/) ) )
44 eleq1 2495 . . . . . . 7  |-  ( ( A  .+  D )  =  (/)  ->  ( ( A  .+  D )  e.  S  <->  (/)  e.  S
) )
4516, 44mtbiri 295 . . . . . 6  |-  ( ( A  .+  D )  =  (/)  ->  -.  ( A  .+  D )  e.  S )
4635simprbi 451 . . . . . . . 8  |-  ( [
<. C ,  D >. ]  .~  =/=  (/)  ->  D  e.  S )
4711caovcl 6233 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  D  e.  S )  ->  ( A  .+  D
)  e.  S )
4847ex 424 . . . . . . . . 9  |-  ( A  e.  S  ->  ( D  e.  S  ->  ( A  .+  D )  e.  S ) )
4948ad2antrr 707 . . . . . . . 8  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( D  e.  S  ->  ( A  .+  D )  e.  S ) )
5046, 49syl5 30 . . . . . . 7  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. C ,  D >. ]  .~  =/=  (/)  ->  ( A  .+  D )  e.  S ) )
5150necon1bd 2666 . . . . . 6  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( -.  ( A  .+  D
)  e.  S  ->  [ <. C ,  D >. ]  .~  =  (/) ) )
5245, 51syl5 30 . . . . 5  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  (
( A  .+  D
)  =  (/)  ->  [ <. C ,  D >. ]  .~  =  (/) ) )
5343, 52impbid 184 . . . 4  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. C ,  D >. ]  .~  =  (/)  <->  ( A  .+  D )  =  (/) ) )
5429, 53syl5bb 249 . . 3  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( (/)  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  (/) ) )
5531eleq2i 2499 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  .~  <->  <. A ,  B >.  e.  ( S  X.  S ) )
56 ecdmn0 6939 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  .~  <->  [ <. A ,  B >. ]  .~  =/=  (/) )
57 opelxp 4900 . . . . . . . 8  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
5855, 56, 573bitr3i 267 . . . . . . 7  |-  ( [
<. A ,  B >. ]  .~  =/=  (/)  <->  ( A  e.  S  /\  B  e.  S ) )
5958simprbi 451 . . . . . 6  |-  ( [
<. A ,  B >. ]  .~  =/=  (/)  ->  B  e.  S )
6059necon1bi 2641 . . . . 5  |-  ( -.  B  e.  S  ->  [ <. A ,  B >. ]  .~  =  (/) )
6160adantl 453 . . . 4  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  [ <. A ,  B >. ]  .~  =  (/) )
6261eqeq1d 2443 . . 3  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  (/)  =  [ <. C ,  D >. ]  .~  ) )
63 simpl 444 . . . . . . 7  |-  ( ( B  e.  S  /\  C  e.  S )  ->  B  e.  S )
6463con3i 129 . . . . . 6  |-  ( -.  B  e.  S  ->  -.  ( B  e.  S  /\  C  e.  S
) )
6515ndmov 6223 . . . . . 6  |-  ( -.  ( B  e.  S  /\  C  e.  S
)  ->  ( B  .+  C )  =  (/) )
6664, 65syl 16 . . . . 5  |-  ( -.  B  e.  S  -> 
( B  .+  C
)  =  (/) )
6766adantl 453 . . . 4  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( B  .+  C )  =  (/) )
6867eqeq2d 2446 . . 3  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  (
( A  .+  D
)  =  ( B 
.+  C )  <->  ( A  .+  D )  =  (/) ) )
6954, 62, 683bitr4d 277 . 2  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
7028, 69pm2.61dan 767 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620   <.cop 3809   class class class wbr 4204    X. cxp 4868   dom cdm 4870  (class class class)co 6073    Er wer 6894   [cec 6895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-ov 6076  df-er 6897  df-ec 6899
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