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Theorem ndmovdistr 7336
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 . . . . . . 7 dom 𝐹 = (𝑆 × 𝑆)
2 ndmov.5 . . . . . . 7 ¬ ∅ ∈ 𝑆
31, 2ndmovrcl 7333 . . . . . 6 ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵𝑆𝐶𝑆))
43anim2i 618 . . . . 5 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
5 3anass 1091 . . . . 5 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
64, 5sylibr 236 . . . 4 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
76con3i 157 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆))
8 ndmov.6 . . . 4 dom 𝐺 = (𝑆 × 𝑆)
98ndmov 7331 . . 3 (¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
107, 9syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
118, 2ndmovrcl 7333 . . . . . 6 ((𝐴𝐺𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
128, 2ndmovrcl 7333 . . . . . 6 ((𝐴𝐺𝐶) ∈ 𝑆 → (𝐴𝑆𝐶𝑆))
1311, 12anim12i 614 . . . . 5 (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝑆𝐵𝑆) ∧ (𝐴𝑆𝐶𝑆)))
14 anandi3 1098 . . . . 5 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ (𝐴𝑆𝐶𝑆)))
1513, 14sylibr 236 . . . 4 (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
1615con3i 157 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆))
171ndmov 7331 . . 3 (¬ ((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
1816, 17syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
1910, 18eqtr4d 2859 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
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:  distrpi  10319  distrnq  10382  distrpr  10449  distrsr  10512
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