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Theorem ecopovsymg 6389
Description: Assuming the operation  F is commutative, show that the relation  .~, specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
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
) ) ) }
ecopoprg.com  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  =  ( y 
.+  x ) )
Assertion
Ref Expression
ecopovsymg  |-  ( 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 ecopovsymg
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 4512 . . . . 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 3056 . . . 4  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
43brel 4490 . . 3  |-  ( A  .~  B  ->  ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S
) ) )
5 eqid 2088 . . . 4  |-  ( S  X.  S )  =  ( S  X.  S
)
6 breq1 3848 . . . . 5  |-  ( <.
f ,  g >.  =  A  ->  ( <.
f ,  g >.  .~  <. h ,  t
>. 
<->  A  .~  <. h ,  t >. )
)
7 breq2 3849 . . . . 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 3849 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( A  .~  <. h ,  t
>. 
<->  A  .~  B ) )
10 breq1 3848 . . . . 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 ) ) )
12 ecopoprg.com . . . . . . . . 9  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  =  ( y 
.+  x ) )
1312adantl 271 . . . . . . . 8  |-  ( ( ( ( f  e.  S  /\  g  e.  S )  /\  (
h  e.  S  /\  t  e.  S )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( y 
.+  x ) )
14 simpll 496 . . . . . . . 8  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
f  e.  S )
15 simprr 499 . . . . . . . 8  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
t  e.  S )
1613, 14, 15caovcomd 5801 . . . . . . 7  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( f  .+  t
)  =  ( t 
.+  f ) )
17 simplr 497 . . . . . . . 8  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
g  e.  S )
18 simprl 498 . . . . . . . 8  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  ->  h  e.  S )
1913, 17, 18caovcomd 5801 . . . . . . 7  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( g  .+  h
)  =  ( h 
.+  g ) )
2016, 19eqeq12d 2102 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( ( f  .+  t )  =  ( g  .+  h )  <-> 
( t  .+  f
)  =  ( h 
.+  g ) ) )
21 eqcom 2090 . . . . . 6  |-  ( ( t  .+  f )  =  ( h  .+  g )  <->  ( h  .+  g )  =  ( t  .+  f ) )
2220, 21syl6bb 194 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( ( f  .+  t )  =  ( g  .+  h )  <-> 
( h  .+  g
)  =  ( t 
.+  f ) ) )
231ecopoveq 6385 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. h ,  t
>. 
<->  ( f  .+  t
)  =  ( g 
.+  h ) ) )
241ecopoveq 6385 . . . . . 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 ) ) )
2622, 23, 253bitr4d 218 . . . 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 4515 . . 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 1289   E.wex 1426    e. wcel 1438   <.cop 3449   class class class wbr 3845   {copab 3898    X. cxp 4436  (class class class)co 5652
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-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  ecopoverg  6391
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