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Theorem ndmovass 7620
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 7618 . . . . 5 ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
43anim1i 615 . . . 4 (((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
5 df-3an 1088 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
64, 5sylibr 234 . . 3 (((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
71ndmov 7616 . . 3 (¬ ((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅)
86, 7nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅)
91, 2ndmovrcl 7618 . . . . 5 ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵𝑆𝐶𝑆))
109anim2i 617 . . . 4 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
11 3anass 1094 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
1210, 11sylibr 234 . . 3 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
131ndmov 7616 . . 3 (¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅)
1412, 13nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅)
158, 14eqtr4d 2777 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  c0 4338   × cxp 5686  dom cdm 5688  (class class class)co 7430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-dm 5698  df-iota 6515  df-fv 6570  df-ov 7433
This theorem is referenced by:  addasspi  10932  mulasspi  10934  addassnq  10995  mulassnq  10996  genpass  11046  addasssr  11125  mulasssr  11127
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