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Theorem ndmovdistr 7326
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 7323 . . . . . 6 ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵𝑆𝐶𝑆))
43anim2i 616 . . . . 5 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
5 3anass 1087 . . . . 5 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
64, 5sylibr 235 . . . 4 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
76con3i 157 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆))
8 ndmov.6 . . . 4 dom 𝐺 = (𝑆 × 𝑆)
98ndmov 7321 . . 3 (¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
107, 9syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
118, 2ndmovrcl 7323 . . . . . 6 ((𝐴𝐺𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
128, 2ndmovrcl 7323 . . . . . 6 ((𝐴𝐺𝐶) ∈ 𝑆 → (𝐴𝑆𝐶𝑆))
1311, 12anim12i 612 . . . . 5 (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝑆𝐵𝑆) ∧ (𝐴𝑆𝐶𝑆)))
14 anandi3 1094 . . . . 5 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ (𝐴𝑆𝐶𝑆)))
1513, 14sylibr 235 . . . 4 (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
1615con3i 157 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆))
171ndmov 7321 . . 3 (¬ ((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
1816, 17syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
1910, 18eqtr4d 2856 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  c0 4288   × cxp 5546  dom cdm 5548  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  distrpi  10308  distrnq  10371  distrpr  10438  distrsr  10501
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