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Theorem ndmovdistr 7622
Description: Any operation is distributive 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
ndmov.6 dom 𝐺 = (𝑆 × 𝑆)
Assertion
Ref Expression
ndmovdistr (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))

Proof of Theorem ndmovdistr
StepHypRef Expression
1 ndmov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
2 ndmov.5 . . . . . 6 ¬ ∅ ∈ 𝑆
31, 2ndmovrcl 7619 . . . . 5 ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵𝑆𝐶𝑆))
43anim2i 617 . . . 4 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
5 3anass 1094 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
64, 5sylibr 234 . . 3 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
7 ndmov.6 . . . 4 dom 𝐺 = (𝑆 × 𝑆)
87ndmov 7617 . . 3 (¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
96, 8nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
107, 2ndmovrcl 7619 . . . . 5 ((𝐴𝐺𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
117, 2ndmovrcl 7619 . . . . 5 ((𝐴𝐺𝐶) ∈ 𝑆 → (𝐴𝑆𝐶𝑆))
1210, 11anim12i 613 . . . 4 (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝑆𝐵𝑆) ∧ (𝐴𝑆𝐶𝑆)))
13 anandi3 1101 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ (𝐴𝑆𝐶𝑆)))
1412, 13sylibr 234 . . 3 (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
151ndmov 7617 . . 3 (¬ ((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
1614, 15nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
179, 16eqtr4d 2778 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  c0 4339   × cxp 5687  dom cdm 5689  (class class class)co 7431
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  distrpi  10936  distrnq  10999  distrpr  11066  distrsr  11129
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