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Theorem ecopover 6204
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 4463 . . . 4  |-  Rel  .~
32a1i 9 . . 3  |-  ( T. 
->  Rel  .~  )
4 ecopopr.com . . . . 5  |-  ( x 
.+  y )  =  ( y  .+  x
)
51, 4ecopovsym 6202 . . . 4  |-  ( f  .~  g  ->  g  .~  f )
65adantl 262 . . 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 6203 . . . 4  |-  ( ( f  .~  g  /\  g  .~  h )  -> 
f  .~  h )
1110adantl 262 . . 3  |-  ( ( T.  /\  ( f  .~  g  /\  g  .~  h ) )  -> 
f  .~  h )
12 vex 2560 . . . . . . . . . . 11  |-  g  e. 
_V
13 vex 2560 . . . . . . . . . . 11  |-  h  e. 
_V
1412, 13, 4caovcom 5658 . . . . . . . . . 10  |-  ( g 
.+  h )  =  ( h  .+  g
)
151ecopoveq 6201 . . . . . . . . . 10  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  -> 
( <. g ,  h >.  .~  <. g ,  h >.  <-> 
( g  .+  h
)  =  ( h 
.+  g ) ) )
1614, 15mpbiri 157 . . . . . . . . 9  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  ->  <. g ,  h >.  .~ 
<. g ,  h >. )
1716anidms 377 . . . . . . . 8  |-  ( ( g  e.  S  /\  h  e.  S )  -> 
<. g ,  h >.  .~ 
<. g ,  h >. )
1817rgen2a 2375 . . . . . . 7  |-  A. g  e.  S  A. h  e.  S  <. g ,  h >.  .~  <. g ,  h >.
19 breq12 3769 . . . . . . . . 9  |-  ( ( f  =  <. g ,  h >.  /\  f  =  <. g ,  h >. )  ->  ( f  .~  f  <->  <. g ,  h >.  .~  <. g ,  h >. ) )
2019anidms 377 . . . . . . . 8  |-  ( f  =  <. g ,  h >.  ->  ( f  .~  f 
<-> 
<. g ,  h >.  .~ 
<. g ,  h >. ) )
2120ralxp 4479 . . . . . . 7  |-  ( A. f  e.  ( S  X.  S ) f  .~  f 
<-> 
A. g  e.  S  A. h  e.  S  <. g ,  h >.  .~ 
<. g ,  h >. )
2218, 21mpbir 134 . . . . . 6  |-  A. f  e.  ( S  X.  S
) f  .~  f
2322rspec 2373 . . . . 5  |-  ( f  e.  ( S  X.  S )  ->  f  .~  f )
2423a1i 9 . . . 4  |-  ( T. 
->  ( f  e.  ( S  X.  S )  ->  f  .~  f
) )
25 opabssxp 4414 . . . . . . 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 2975 . . . . . 6  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
2726ssbri 3806 . . . . 5  |-  ( f  .~  f  ->  f
( ( S  X.  S )  X.  ( S  X.  S ) ) f )
28 brxp 4375 . . . . . 6  |-  ( f ( ( S  X.  S )  X.  ( S  X.  S ) ) f  <->  ( f  e.  ( S  X.  S
)  /\  f  e.  ( S  X.  S
) ) )
2928simplbi 259 . . . . 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 130 . . 3  |-  ( T. 
->  ( f  e.  ( S  X.  S )  <-> 
f  .~  f )
)
323, 6, 11, 31iserd 6132 . 2  |-  ( T. 
->  .~  Er  ( S  X.  S ) )
3332trud 1252 1  |-  .~  Er  ( S  X.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243   T. wtru 1244   E.wex 1381    e. wcel 1393   A.wral 2306   <.cop 3378   class class class wbr 3764   {copab 3817    X. cxp 4343   Rel wrel 4350  (class class class)co 5512    Er wer 6103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fv 4910  df-ov 5515  df-er 6106
This theorem is referenced by: (None)
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