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Theorem ecopover 6388
Description: Assuming that operation  F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation  .~, specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ecopopr.1  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z 
.+  u )  =  ( w  .+  v
) ) ) }
ecopopr.com  |-  ( x 
.+  y )  =  ( y  .+  x
)
ecopopr.cl  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
ecopopr.ass  |-  ( ( x  .+  y ) 
.+  z )  =  ( x  .+  (
y  .+  z )
)
ecopopr.can  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
Assertion
Ref Expression
ecopover  |-  .~  Er  ( S  X.  S
)
Distinct variable groups:    x, y, z, w, v, u,  .+    x, S, y, z, w, v, u
Allowed substitution hints:    .~ ( x, y,
z, w, v, u)

Proof of Theorem ecopover
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z 
.+  u )  =  ( w  .+  v
) ) ) }
21relopabi 4563 . . . 4  |-  Rel  .~
32a1i 9 . . 3  |-  ( T. 
->  Rel  .~  )
4 ecopopr.com . . . . 5  |-  ( x 
.+  y )  =  ( y  .+  x
)
51, 4ecopovsym 6386 . . . 4  |-  ( f  .~  g  ->  g  .~  f )
65adantl 271 . . 3  |-  ( ( T.  /\  f  .~  g )  ->  g  .~  f )
7 ecopopr.cl . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
8 ecopopr.ass . . . . 5  |-  ( ( x  .+  y ) 
.+  z )  =  ( x  .+  (
y  .+  z )
)
9 ecopopr.can . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
101, 4, 7, 8, 9ecopovtrn 6387 . . . 4  |-  ( ( f  .~  g  /\  g  .~  h )  -> 
f  .~  h )
1110adantl 271 . . 3  |-  ( ( T.  /\  ( f  .~  g  /\  g  .~  h ) )  -> 
f  .~  h )
12 vex 2622 . . . . . . . . . . 11  |-  g  e. 
_V
13 vex 2622 . . . . . . . . . . 11  |-  h  e. 
_V
1412, 13, 4caovcom 5802 . . . . . . . . . 10  |-  ( g 
.+  h )  =  ( h  .+  g
)
151ecopoveq 6385 . . . . . . . . . 10  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  -> 
( <. g ,  h >.  .~  <. g ,  h >.  <-> 
( g  .+  h
)  =  ( h 
.+  g ) ) )
1614, 15mpbiri 166 . . . . . . . . 9  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  ->  <. g ,  h >.  .~ 
<. g ,  h >. )
1716anidms 389 . . . . . . . 8  |-  ( ( g  e.  S  /\  h  e.  S )  -> 
<. g ,  h >.  .~ 
<. g ,  h >. )
1817rgen2a 2429 . . . . . . 7  |-  A. g  e.  S  A. h  e.  S  <. g ,  h >.  .~  <. g ,  h >.
19 breq12 3850 . . . . . . . . 9  |-  ( ( f  =  <. g ,  h >.  /\  f  =  <. g ,  h >. )  ->  ( f  .~  f  <->  <. g ,  h >.  .~  <. g ,  h >. ) )
2019anidms 389 . . . . . . . 8  |-  ( f  =  <. g ,  h >.  ->  ( f  .~  f 
<-> 
<. g ,  h >.  .~ 
<. g ,  h >. ) )
2120ralxp 4579 . . . . . . 7  |-  ( A. f  e.  ( S  X.  S ) f  .~  f 
<-> 
A. g  e.  S  A. h  e.  S  <. g ,  h >.  .~ 
<. g ,  h >. )
2218, 21mpbir 144 . . . . . 6  |-  A. f  e.  ( S  X.  S
) f  .~  f
2322rspec 2427 . . . . 5  |-  ( f  e.  ( S  X.  S )  ->  f  .~  f )
2423a1i 9 . . . 4  |-  ( T. 
->  ( f  e.  ( S  X.  S )  ->  f  .~  f
) )
25 opabssxp 4512 . . . . . . 7  |-  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .+  u
)  =  ( w 
.+  v ) ) ) }  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
261, 25eqsstri 3056 . . . . . 6  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
2726ssbri 3887 . . . . 5  |-  ( f  .~  f  ->  f
( ( S  X.  S )  X.  ( S  X.  S ) ) f )
28 brxp 4468 . . . . . 6  |-  ( f ( ( S  X.  S )  X.  ( S  X.  S ) ) f  <->  ( f  e.  ( S  X.  S
)  /\  f  e.  ( S  X.  S
) ) )
2928simplbi 268 . . . . 5  |-  ( f ( ( S  X.  S )  X.  ( S  X.  S ) ) f  ->  f  e.  ( S  X.  S
) )
3027, 29syl 14 . . . 4  |-  ( f  .~  f  ->  f  e.  ( S  X.  S
) )
3124, 30impbid1 140 . . 3  |-  ( T. 
->  ( f  e.  ( S  X.  S )  <-> 
f  .~  f )
)
323, 6, 11, 31iserd 6316 . 2  |-  ( T. 
->  .~  Er  ( S  X.  S ) )
3332mptru 1298 1  |-  .~  Er  ( S  X.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   T. wtru 1290   E.wex 1426    e. wcel 1438   A.wral 2359   <.cop 3449   class class class wbr 3845   {copab 3898    X. cxp 4436   Rel wrel 4443  (class class class)co 5652    Er wer 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fv 5023  df-ov 5655  df-er 6290
This theorem is referenced by: (None)
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