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

Theorem eceqoveq 6946
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 4851 . . . . . . . 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 6885 . . . . . . 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 4853 . . . . . 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 6181 . . . . . . 7  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( B  .+  C
)  e.  S )
13 eleq1 2448 . . . . . . 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 6173 . . . . . . 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 6886 . . . . . 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 2390 . . . 4  |-  ( (/)  =  [ <. C ,  D >. ]  .~  <->  [ <. C ,  D >. ]  .~  =  (/) )
30 erdm 6852 . . . . . . . . . . . 12  |-  (  .~  Er  ( S  X.  S
)  ->  dom  .~  =  ( S  X.  S
) )
313, 30ax-mp 8 . . . . . . . . . . 11  |-  dom  .~  =  ( S  X.  S )
3231eleq2i 2452 . . . . . . . . . 10  |-  ( <. C ,  D >.  e. 
dom  .~  <->  <. C ,  D >.  e.  ( S  X.  S ) )
33 ecdmn0 6884 . . . . . . . . . 10  |-  ( <. C ,  D >.  e. 
dom  .~  <->  [ <. C ,  D >. ]  .~  =/=  (/) )
34 opelxp 4849 . . . . . . . . . 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 2600 . . . . . 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 6171 . . . . . . 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 2448 . . . . . . 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 6181 . . . . . . . . . 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 2619 . . . . . 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 2452 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  .~  <->  <. A ,  B >.  e.  ( S  X.  S ) )
56 ecdmn0 6884 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  .~  <->  [ <. A ,  B >. ]  .~  =/=  (/) )
57 opelxp 4849 . . . . . . . 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 2594 . . . . 5  |-  ( -.  B  e.  S  ->  [ <. A ,  B >. ]  .~  =  (/) )
6160adantl 453 . . . 4  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  -.  B  e.  S )  ->  [ <. A ,  B >. ]  .~  =  (/) )
6261eqeq1d 2396 . . 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 6171 . . . . . 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 2399 . . 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 1649    e. wcel 1717    =/= wne 2551   (/)c0 3572   <.cop 3761   class class class wbr 4154    X. cxp 4817   dom cdm 4819  (class class class)co 6021    Er wer 6839   [cec 6840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fv 5403  df-ov 6024  df-er 6842  df-ec 6844
  Copyright terms: Public domain W3C validator