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Theorem ecopover 6967
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 4959 . . . 4  |-  Rel  .~
32a1i 11 . . 3  |-  (  T. 
->  Rel  .~  )
4 ecopopr.com . . . . 5  |-  ( x 
.+  y )  =  ( y  .+  x
)
51, 4ecopovsym 6965 . . . 4  |-  ( f  .~  g  ->  g  .~  f )
65adantl 453 . . 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 6966 . . . 4  |-  ( ( f  .~  g  /\  g  .~  h )  -> 
f  .~  h )
1110adantl 453 . . 3  |-  ( (  T.  /\  ( f  .~  g  /\  g  .~  h ) )  -> 
f  .~  h )
12 vex 2919 . . . . . . . . . . 11  |-  g  e. 
_V
13 vex 2919 . . . . . . . . . . 11  |-  h  e. 
_V
1412, 13, 4caovcom 6203 . . . . . . . . . 10  |-  ( g 
.+  h )  =  ( h  .+  g
)
151ecopoveq 6964 . . . . . . . . . 10  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  -> 
( <. g ,  h >.  .~  <. g ,  h >.  <-> 
( g  .+  h
)  =  ( h 
.+  g ) ) )
1614, 15mpbiri 225 . . . . . . . . 9  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  ->  <. g ,  h >.  .~ 
<. g ,  h >. )
1716anidms 627 . . . . . . . 8  |-  ( ( g  e.  S  /\  h  e.  S )  -> 
<. g ,  h >.  .~ 
<. g ,  h >. )
1817rgen2a 2732 . . . . . . 7  |-  A. g  e.  S  A. h  e.  S  <. g ,  h >.  .~  <. g ,  h >.
19 breq12 4177 . . . . . . . . 9  |-  ( ( f  =  <. g ,  h >.  /\  f  =  <. g ,  h >. )  ->  ( f  .~  f  <->  <. g ,  h >.  .~  <. g ,  h >. ) )
2019anidms 627 . . . . . . . 8  |-  ( f  =  <. g ,  h >.  ->  ( f  .~  f 
<-> 
<. g ,  h >.  .~ 
<. g ,  h >. ) )
2120ralxp 4975 . . . . . . 7  |-  ( A. f  e.  ( S  X.  S ) f  .~  f 
<-> 
A. g  e.  S  A. h  e.  S  <. g ,  h >.  .~ 
<. g ,  h >. )
2218, 21mpbir 201 . . . . . 6  |-  A. f  e.  ( S  X.  S
) f  .~  f
2322rspec 2730 . . . . 5  |-  ( f  e.  ( S  X.  S )  ->  f  .~  f )
2423a1i 11 . . . 4  |-  (  T. 
->  ( f  e.  ( S  X.  S )  ->  f  .~  f
) )
25 opabssxp 4909 . . . . . . 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 3338 . . . . . 6  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
2726ssbri 4214 . . . . 5  |-  ( f  .~  f  ->  f
( ( S  X.  S )  X.  ( S  X.  S ) ) f )
28 brxp 4868 . . . . . 6  |-  ( f ( ( S  X.  S )  X.  ( S  X.  S ) ) f  <->  ( f  e.  ( S  X.  S
)  /\  f  e.  ( S  X.  S
) ) )
2928simplbi 447 . . . . 5  |-  ( f ( ( S  X.  S )  X.  ( S  X.  S ) ) f  ->  f  e.  ( S  X.  S
) )
3027, 29syl 16 . . . 4  |-  ( f  .~  f  ->  f  e.  ( S  X.  S
) )
3124, 30impbid1 195 . . 3  |-  (  T. 
->  ( f  e.  ( S  X.  S )  <-> 
f  .~  f )
)
323, 6, 11, 31iserd 6890 . 2  |-  (  T. 
->  .~  Er  ( S  X.  S ) )
3332trud 1329 1  |-  .~  Er  ( S  X.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    T. wtru 1322   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   <.cop 3777   class class class wbr 4172   {copab 4225    X. cxp 4835   Rel wrel 4842  (class class class)co 6040    Er wer 6861
This theorem is referenced by:  enqer  8754  enrer  8899
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fv 5421  df-ov 6043  df-er 6864
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