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Theorem ndmovass 7547
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 7545 . . . . 5 ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
43anim1i 616 . . . 4 (((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
5 df-3an 1090 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
64, 5sylibr 233 . . 3 (((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
71ndmov 7543 . . 3 (¬ ((𝐴𝐹𝐵) ∈ 𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅)
86, 7nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅)
91, 2ndmovrcl 7545 . . . . 5 ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵𝑆𝐶𝑆))
109anim2i 618 . . . 4 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
11 3anass 1096 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
1210, 11sylibr 233 . . 3 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
131ndmov 7543 . . 3 (¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅)
1412, 13nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅)
158, 14eqtr4d 2780 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  c0 4287   × cxp 5636  dom cdm 5638  (class class class)co 7362
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-dm 5648  df-iota 6453  df-fv 6509  df-ov 7365
This theorem is referenced by:  addasspi  10838  mulasspi  10840  addassnq  10901  mulassnq  10902  genpass  10952  addasssr  11031  mulasssr  11033
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