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Theorem ndmovcom 6435
Description: Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
Hypothesis
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
Assertion
Ref Expression
ndmovcom  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  ( B F A ) )

Proof of Theorem ndmovcom
StepHypRef Expression
1 ndmov.1 . . 3  |-  dom  F  =  ( S  X.  S )
21ndmov 6432 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
3 ancom 448 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  ( B  e.  S  /\  A  e.  S )
)
41ndmov 6432 . . 3  |-  ( -.  ( B  e.  S  /\  A  e.  S
)  ->  ( B F A )  =  (/) )
53, 4sylnbi 304 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( B F A )  =  (/) )
62, 5eqtr4d 2498 1  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  ( B F A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   (/)c0 3783    X. cxp 4986   dom cdm 4988  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-dm 4998  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  addcompi  9261  mulcompi  9263  addcompq  9317  addcomnq  9318  mulcompq  9319  mulcomnq  9320  addcompr  9388  mulcompr  9390  addcomsr  9453  mulcomsr  9455  addcomgi  31606
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