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Theorem ndmovass 7335
 Description: Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
Hypotheses
Ref Expression
ndmov.1 dom 𝐹 = (𝑆 × 𝑆)
ndmov.5 ¬ ∅ ∈ 𝑆
Assertion
Ref Expression
ndmovass (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Proof of Theorem ndmovass
StepHypRef Expression
1 ndmov.1 . . . . . . 7 dom 𝐹 = (𝑆 × 𝑆)
2 ndmov.5 . . . . . . 7 ¬ ∅ ∈ 𝑆
31, 2ndmovrcl 7333 . . . . . 6 ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
43anim1i 616 . . . . 5 (((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
5 df-3an 1085 . . . . 5 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
64, 5sylibr 236 . . . 4 (((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
76con3i 157 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆))
81ndmov 7331 . . 3 (¬ ((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅)
97, 8syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅)
101, 2ndmovrcl 7333 . . . . . 6 ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵𝑆𝐶𝑆))
1110anim2i 618 . . . . 5 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
12 3anass 1091 . . . . 5 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
1311, 12sylibr 236 . . . 4 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
1413con3i 157 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆))
151ndmov 7331 . . 3 (¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅)
1614, 15syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅)
179, 16eqtr4d 2859 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∅c0 4290   × cxp 5552  dom cdm 5554  (class class class)co 7155 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-xp 5560  df-dm 5564  df-iota 6313  df-fv 6362  df-ov 7158 This theorem is referenced by:  addasspi  10316  mulasspi  10318  addassnq  10379  mulassnq  10380  genpass  10430  addasssr  10509  mulasssr  10511
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