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Theorem ndmovass 7460
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 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
2 ndmov.5 . . . . . 6 ¬ ∅ ∈ 𝑆
31, 2ndmovrcl 7458 . . . . 5 ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
43anim1i 615 . . . 4 (((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
5 df-3an 1088 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
64, 5sylibr 233 . . 3 (((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
71ndmov 7456 . . 3 (¬ ((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅)
86, 7nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅)
91, 2ndmovrcl 7458 . . . . 5 ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵𝑆𝐶𝑆))
109anim2i 617 . . . 4 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
11 3anass 1094 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
1210, 11sylibr 233 . . 3 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
131ndmov 7456 . . 3 (¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅)
1412, 13nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅)
158, 14eqtr4d 2781 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  c0 4256   × cxp 5587  dom cdm 5589  (class class class)co 7275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599  df-iota 6391  df-fv 6441  df-ov 7278
This theorem is referenced by:  addasspi  10651  mulasspi  10653  addassnq  10714  mulassnq  10715  genpass  10765  addasssr  10844  mulasssr  10846
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