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Theorem ndmaovcom 28045
Description: Any operation is commutative outside its domain, analogous to ndmovcom 6234. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
ndmaov.1  |-  dom  F  =  ( S  X.  S )
Assertion
Ref Expression
ndmaovcom  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( A F B))  = (( B F A))  )

Proof of Theorem ndmaovcom
StepHypRef Expression
1 opelxp 4908 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
2 ndmaov.1 . . . . . 6  |-  dom  F  =  ( S  X.  S )
32eqcomi 2440 . . . . 5  |-  ( S  X.  S )  =  dom  F
43eleq2i 2500 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  <. A ,  B >.  e.  dom  F )
51, 4bitr3i 243 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  <. A ,  B >.  e. 
dom  F )
6 ndmaov 28023 . . 3  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  =  _V )
75, 6sylnbi 298 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( A F B))  =  _V )
8 ancom 438 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  <->  ( B  e.  S  /\  A  e.  S )
)
9 opelxp 4908 . . . 4  |-  ( <. B ,  A >.  e.  ( S  X.  S
)  <->  ( B  e.  S  /\  A  e.  S ) )
103eleq2i 2500 . . . 4  |-  ( <. B ,  A >.  e.  ( S  X.  S
)  <->  <. B ,  A >.  e.  dom  F )
118, 9, 103bitr2i 265 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  <. B ,  A >.  e. 
dom  F )
12 ndmaov 28023 . . 3  |-  ( -. 
<. B ,  A >.  e. 
dom  F  -> (( B F A))  =  _V )
1311, 12sylnbi 298 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( B F A))  =  _V )
147, 13eqtr4d 2471 1  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( A F B))  = (( B F A))  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    X. cxp 4876   dom cdm 4878   ((caov 27949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884  df-fv 5462  df-dfat 27950  df-afv 27951  df-aov 27952
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