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Theorem ndmovcom 6775
Description: Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
Hypothesis
Ref Expression
ndmov.1 dom 𝐹 = (𝑆 × 𝑆)
Assertion
Ref Expression
ndmovcom (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Proof of Theorem ndmovcom
StepHypRef Expression
1 ndmov.1 . . 3 dom 𝐹 = (𝑆 × 𝑆)
21ndmov 6772 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
3 ancom 466 . . 3 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
41ndmov 6772 . . 3 (¬ (𝐵𝑆𝐴𝑆) → (𝐵𝐹𝐴) = ∅)
53, 4sylnbi 320 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐵𝐹𝐴) = ∅)
62, 5eqtr4d 2663 1 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1992  c0 3896   × cxp 5077  dom cdm 5079  (class class class)co 6605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-dm 5089  df-iota 5813  df-fv 5858  df-ov 6608
This theorem is referenced by:  addcompi  9661  mulcompi  9663  addcompq  9717  addcomnq  9718  mulcompq  9719  mulcomnq  9720  addcompr  9788  mulcompr  9790  addcomsr  9853  mulcomsr  9855  addcomgi  38128
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