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Theorem ndmovcom 6447
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 6444 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
3 ancom 450 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  ( B  e.  S  /\  A  e.  S )
)
41ndmov 6444 . . 3  |-  ( -.  ( B  e.  S  /\  A  e.  S
)  ->  ( B F A )  =  (/) )
53, 4sylnbi 306 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( B F A )  =  (/) )
62, 5eqtr4d 2487 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 369    = wceq 1383    e. wcel 1804   (/)c0 3770    X. cxp 4987   dom cdm 4989  (class class class)co 6281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-dm 4999  df-iota 5541  df-fv 5586  df-ov 6284
This theorem is referenced by:  addcompi  9275  mulcompi  9277  addcompq  9331  addcomnq  9332  mulcompq  9333  mulcomnq  9334  addcompr  9402  mulcompr  9404  addcomsr  9467  mulcomsr  9469  addcomgi  31319
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