Theorem ndmovdistr 7606

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	⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))

Proof of Theorem ndmovdistr

Step	Hyp	Ref	Expression
1		ndmov.1	. . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆)
2		ndmov.5	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑆
3	1, 2	ndmovrcl 7603	. . . . 5 ⊢ ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
4	3	anim2i 629	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
5		3anass 1111	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
6	4, 5	sylibr 237	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
7		ndmov.6	. . . 4 ⊢ dom 𝐺 = (𝑆 × 𝑆)
8	7	ndmov 7601	. . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
9	6, 8	nsyl5 160	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
10	7, 2	ndmovrcl 7603	. . . . 5 ⊢ ((𝐴𝐺𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
11	7, 2	ndmovrcl 7603	. . . . 5 ⊢ ((𝐴𝐺𝐶) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
12	10, 11	anim12i 625	. . . 4 ⊢ (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
13		anandi3 1119	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
14	12, 13	sylibr 237	. . 3 ⊢ (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
15	1	ndmov 7601	. . 3 ⊢ (¬ ((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
16	14, 15	nsyl5 160	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
17	9, 16	eqtr4d 2800	1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∅c0 4282   × cxp 5657  dom cdm 5659  (class class class)co 7416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7419

This theorem is used by:  distrpi  10910  distrnq  10973  distrpr  11040  distrsr  11103