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Theorem ndmovdistr 7593
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 7590 . . . . 5 ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵𝑆𝐶𝑆))
43anim2i 616 . . . 4 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
5 3anass 1092 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
64, 5sylibr 233 . . 3 ((𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
7 ndmov.6 . . . 4 dom 𝐺 = (𝑆 × 𝑆)
87ndmov 7588 . . 3 (¬ (𝐴𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
96, 8nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
107, 2ndmovrcl 7590 . . . . 5 ((𝐴𝐺𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
117, 2ndmovrcl 7590 . . . . 5 ((𝐴𝐺𝐶) ∈ 𝑆 → (𝐴𝑆𝐶𝑆))
1210, 11anim12i 612 . . . 4 (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝑆𝐵𝑆) ∧ (𝐴𝑆𝐶𝑆)))
13 anandi3 1099 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ (𝐴𝑆𝐶𝑆)))
1412, 13sylibr 233 . . 3 (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
151ndmov 7588 . . 3 (¬ ((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
1614, 15nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
179, 16eqtr4d 2769 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  c0 4317   × cxp 5667  dom cdm 5669  (class class class)co 7405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-iota 6489  df-fv 6545  df-ov 7408
This theorem is referenced by:  distrpi  10895  distrnq  10958  distrpr  11025  distrsr  11088
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