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Theorem ndmovcom 6986
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 6983 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
3 ancom 465 . . 3 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
41ndmov 6983 . . 3 (¬ (𝐵𝑆𝐴𝑆) → (𝐵𝐹𝐴) = ∅)
53, 4sylnbi 319 . 2 (¬ (𝐴𝑆𝐵𝑆) → (𝐵𝐹𝐴) = ∅)
62, 5eqtr4d 2797 1 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  c0 4058   × cxp 5264  dom cdm 5266  (class class class)co 6813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-dm 5276  df-iota 6012  df-fv 6057  df-ov 6816
This theorem is referenced by:  addcompi  9908  mulcompi  9910  addcompq  9964  addcomnq  9965  mulcompq  9966  mulcomnq  9967  addcompr  10035  mulcompr  10037  addcomsr  10100  mulcomsr  10102  addcomgi  39162
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