Theorem ndmovdistr 7549

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 7546	. . . . 5 ⊢ ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
4	3	anim2i 624	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
5		3anass 1101	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
6	4, 5	sylibr 236	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
7		ndmov.6	. . . 4 ⊢ dom 𝐺 = (𝑆 × 𝑆)
8	7	ndmov 7544	. . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
9	6, 8	nsyl5 159	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ∅)
10	7, 2	ndmovrcl 7546	. . . . 5 ⊢ ((𝐴𝐺𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
11	7, 2	ndmovrcl 7546	. . . . 5 ⊢ ((𝐴𝐺𝐶) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
12	10, 11	anim12i 620	. . . 4 ⊢ (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
13		anandi3 1108	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
14	12, 13	sylibr 236	. . 3 ⊢ (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
15	1	ndmov 7544	. . 3 ⊢ (¬ ((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
16	14, 15	nsyl5 159	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) = ∅)
17	9, 16	eqtr4d 2779	1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∅c0 4264   × cxp 5619  dom cdm 5621  (class class class)co 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-xp 5627  df-dm 5631  df-iota 6445  df-fv 6497  df-ov 7363

This theorem is referenced by:  distrpi  10816  distrnq  10879  distrpr  10946  distrsr  11009