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Theorem ndmovcom 7620
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 7617 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
3 ancom 460 . . 3 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
41ndmov 7617 . . 3 (¬ (𝐵𝑆𝐴𝑆) → (𝐵𝐹𝐴) = ∅)
53, 4sylnbi 330 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐵𝐹𝐴) = ∅)
62, 5eqtr4d 2778 1 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  c0 4339   × cxp 5687  dom cdm 5689  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  addcompi  10932  mulcompi  10934  addcompq  10988  addcomnq  10989  mulcompq  10990  mulcomnq  10991  addcompr  11059  mulcompr  11061  addcomsr  11125  mulcomsr  11127  addcomgi  44452
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