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Theorem ecopovsym 7009
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 4953 . . . . 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 3380 . . . 4  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
43brel 4929 . . 3  |-  ( A  .~  B  ->  ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S
) ) )
5 eqid 2438 . . . 4  |-  ( S  X.  S )  =  ( S  X.  S
)
6 breq1 4218 . . . . 5  |-  ( <.
f ,  g >.  =  A  ->  ( <.
f ,  g >.  .~  <. h ,  t
>. 
<->  A  .~  <. h ,  t >. )
)
7 breq2 4219 . . . . 5  |-  ( <.
f ,  g >.  =  A  ->  ( <.
h ,  t >.  .~  <. f ,  g
>. 
<-> 
<. h ,  t >.  .~  A ) )
86, 7bibi12d 314 . . . 4  |-  ( <.
f ,  g >.  =  A  ->  ( (
<. f ,  g >.  .~  <. h ,  t
>. 
<-> 
<. h ,  t >.  .~  <. f ,  g
>. )  <->  ( A  .~  <.
h ,  t >.  <->  <.
h ,  t >.  .~  A ) ) )
9 breq2 4219 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( A  .~  <. h ,  t
>. 
<->  A  .~  B ) )
10 breq1 4218 . . . . 5  |-  ( <.
h ,  t >.  =  B  ->  ( <.
h ,  t >.  .~  A  <->  B  .~  A ) )
119, 10bibi12d 314 . . . 4  |-  ( <.
h ,  t >.  =  B  ->  ( ( A  .~  <. h ,  t >.  <->  <. h ,  t >.  .~  A )  <-> 
( A  .~  B  <->  B  .~  A ) ) )
121ecopoveq 7008 . . . . . 6  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. h ,  t
>. 
<->  ( f  .+  t
)  =  ( g 
.+  h ) ) )
13 vex 2961 . . . . . . . . 9  |-  f  e. 
_V
14 vex 2961 . . . . . . . . 9  |-  t  e. 
_V
15 ecopopr.com . . . . . . . . 9  |-  ( x 
.+  y )  =  ( y  .+  x
)
1613, 14, 15caovcom 6247 . . . . . . . 8  |-  ( f 
.+  t )  =  ( t  .+  f
)
17 vex 2961 . . . . . . . . 9  |-  g  e. 
_V
18 vex 2961 . . . . . . . . 9  |-  h  e. 
_V
1917, 18, 15caovcom 6247 . . . . . . . 8  |-  ( g 
.+  h )  =  ( h  .+  g
)
2016, 19eqeq12i 2451 . . . . . . 7  |-  ( ( f  .+  t )  =  ( g  .+  h )  <->  ( t  .+  f )  =  ( h  .+  g ) )
21 eqcom 2440 . . . . . . 7  |-  ( ( t  .+  f )  =  ( h  .+  g )  <->  ( h  .+  g )  =  ( t  .+  f ) )
2220, 21bitri 242 . . . . . 6  |-  ( ( f  .+  t )  =  ( g  .+  h )  <->  ( h  .+  g )  =  ( t  .+  f ) )
2312, 22syl6bb 254 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. f ,  g
>.  .~  <. h ,  t
>. 
<->  ( h  .+  g
)  =  ( t 
.+  f ) ) )
241ecopoveq 7008 . . . . . 6  |-  ( ( ( h  e.  S  /\  t  e.  S
)  /\  ( f  e.  S  /\  g  e.  S ) )  -> 
( <. h ,  t
>.  .~  <. f ,  g
>. 
<->  ( h  .+  g
)  =  ( t 
.+  f ) ) )
2524ancoms 441 . . . . 5  |-  ( ( ( f  e.  S  /\  g  e.  S
)  /\  ( h  e.  S  /\  t  e.  S ) )  -> 
( <. h ,  t
>.  .~  <. f ,  g
>. 
<->  ( h  .+  g
)  =  ( t 
.+  f ) ) )
2623, 25bitr4d 249 . . . 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 4956 . . 3  |-  ( ( A  e.  ( S  X.  S )  /\  B  e.  ( S  X.  S ) )  -> 
( A  .~  B  <->  B  .~  A ) )
284, 27syl 16 . 2  |-  ( A  .~  B  ->  ( A  .~  B  <->  B  .~  A ) )
2928ibi 234 1  |-  ( A  .~  B  ->  B  .~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4215   {copab 4268    X. cxp 4879  (class class class)co 6084
This theorem is referenced by:  ecopover  7011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-xp 4887  df-iota 5421  df-fv 5465  df-ov 6087
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