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Theorem ecopovsym 6268
Description: Assuming the operation  F is commutative, show that the relation  .~, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-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
)
Assertion
Ref Expression
ecopovsym  |-  ( A  .~  B  ->  B  .~  A )
Distinct variable groups:    x, y, z, w, v, u,  .+    x, S, y, z, w, v, u
Allowed substitution hints:    A( x, y, z, w, v, u)    B( x, y, z, w, v, u)    .~ ( x, y, z, w, v, u)

Proof of Theorem ecopovsym
Dummy variables  f  g  h  t 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
) ) ) }
2 opabssxp 4440 . . . . 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 ) ) ) }  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
31, 2eqsstri 3030 . . . 4  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
43brel 4418 . . 3  |-  ( A  .~  B  ->  ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S
) ) )
5 eqid 2082 . . . 4  |-  ( S  X.  S )  =  ( S  X.  S
)
6 breq1 3796 . . . . 5  |-  ( <.
f ,  g >.  =  A  ->  ( <.
f ,  g >.  .~  <. h ,  t
>. 
<->  A  .~  <. h ,  t >. )
)
7 breq2 3797 . . . . 5  |-  ( <.
f ,  g >.  =  A  ->  ( <.
h ,  t >.  .~  <. f ,  g
>. 
<-> 
<. h ,  t >.  .~  A ) )
86, 7bibi12d 233 . . . 4  |-  ( <.
f ,  g >.  =  A  ->  ( (
<. f ,  g >.  .~  <. h ,  t
>. 
<-> 
<. h ,  t >.  .~  <. f ,  g
>. )  <->  ( A  .~  <.
h ,  t >.  <->  <.
h ,  t >.  .~  A ) ) )
9 breq2 3797 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( A  .~  <. h ,  t
>. 
<->  A  .~  B ) )
10 breq1 3796 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( <.
h ,  t >.  .~  A  <->  B  .~  A ) )
119, 10bibi12d 233 . . . 4  |-  ( <.
h ,  t >.  =  B  ->  ( ( A  .~  <. h ,  t >.  <->  <. h ,  t >.  .~  A )  <-> 
( A  .~  B  <->  B  .~  A ) ) )
121ecopoveq 6267 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. h ,  t
>. 
<->  ( f  .+  t
)  =  ( g 
.+  h ) ) )
13 vex 2605 . . . . . . . . 9  |-  f  e. 
_V
14 vex 2605 . . . . . . . . 9  |-  t  e. 
_V
15 ecopopr.com . . . . . . . . 9  |-  ( x 
.+  y )  =  ( y  .+  x
)
1613, 14, 15caovcom 5689 . . . . . . . 8  |-  ( f 
.+  t )  =  ( t  .+  f
)
17 vex 2605 . . . . . . . . 9  |-  g  e. 
_V
18 vex 2605 . . . . . . . . 9  |-  h  e. 
_V
1917, 18, 15caovcom 5689 . . . . . . . 8  |-  ( g 
.+  h )  =  ( h  .+  g
)
2016, 19eqeq12i 2095 . . . . . . 7  |-  ( ( f  .+  t )  =  ( g  .+  h )  <->  ( t  .+  f )  =  ( h  .+  g ) )
21 eqcom 2084 . . . . . . 7  |-  ( ( t  .+  f )  =  ( h  .+  g )  <->  ( h  .+  g )  =  ( t  .+  f ) )
2220, 21bitri 182 . . . . . 6  |-  ( ( f  .+  t )  =  ( g  .+  h )  <->  ( h  .+  g )  =  ( t  .+  f ) )
2312, 22syl6bb 194 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. h ,  t
>. 
<->  ( h  .+  g
)  =  ( t 
.+  f ) ) )
241ecopoveq 6267 . . . . . 6  |-  ( ( ( h  e.  S  /\  t  e.  S
)  /\  ( f  e.  S  /\  g  e.  S ) )  -> 
( <. h ,  t
>.  .~  <. f ,  g
>. 
<->  ( h  .+  g
)  =  ( t 
.+  f ) ) )
2524ancoms 264 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. h ,  t
>.  .~  <. f ,  g
>. 
<->  ( h  .+  g
)  =  ( t 
.+  f ) ) )
2623, 25bitr4d 189 . . . 4  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. h ,  t
>. 
<-> 
<. h ,  t >.  .~  <. f ,  g
>. ) )
275, 8, 11, 262optocl 4443 . . 3  |-  ( ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S ) )  -> 
( A  .~  B  <->  B  .~  A ) )
284, 27syl 14 . 2  |-  ( A  .~  B  ->  ( A  .~  B  <->  B  .~  A ) )
2928ibi 174 1  |-  ( A  .~  B  ->  B  .~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   <.cop 3409   class class class wbr 3793   {copab 3846    X. cxp 4369  (class class class)co 5543
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  ecopover  6270
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