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Theorem ndmovcom 6719
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 6716 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
3 ancom 465 . . 3 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
41ndmov 6716 . . 3 (¬ (𝐵𝑆𝐴𝑆) → (𝐵𝐹𝐴) = ∅)
53, 4sylnbi 319 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐵𝐹𝐴) = ∅)
62, 5eqtr4d 2647 1 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  c0 3874   × cxp 5036  dom cdm 5038  (class class class)co 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552
This theorem is referenced by:  addcompi  9595  mulcompi  9597  addcompq  9651  addcomnq  9652  mulcompq  9653  mulcomnq  9654  addcompr  9722  mulcompr  9724  addcomsr  9787  mulcomsr  9789  addcomgi  37681
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