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Theorem ndmovcom 6863
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 6860 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
3 ancom 465 . . 3 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
41ndmov 6860 . . 3 (¬ (𝐵𝑆𝐴𝑆) → (𝐵𝐹𝐴) = ∅)
53, 4sylnbi 319 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐵𝐹𝐴) = ∅)
62, 5eqtr4d 2688 1 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  c0 3948   × cxp 5141  dom cdm 5143  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  addcompi  9754  mulcompi  9756  addcompq  9810  addcomnq  9811  mulcompq  9812  mulcomnq  9813  addcompr  9881  mulcompr  9883  addcomsr  9946  mulcomsr  9948  addcomgi  38977
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