Theorem ndmovdistr 7639

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 7636	. . . . 5 ⊢ ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
4	3	anim2i 616	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
5		3anass 1095	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
6	4, 5	sylibr 234	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
7		ndmov.6	. . . 4 ⊢ dom 𝐺 = (𝑆 × 𝑆)
8	7	ndmov 7634	. . 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 7636	. . . . 5 ⊢ ((𝐴𝐺𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
11	7, 2	ndmovrcl 7636	. . . . 5 ⊢ ((𝐴𝐺𝐶) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
12	10, 11	anim12i 612	. . . 4 ⊢ (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
13		anandi3 1102	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
14	12, 13	sylibr 234	. . 3 ⊢ (((𝐴𝐺𝐵) ∈ 𝑆 ∧ (𝐴𝐺𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
15	1	ndmov 7634	. . 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 2783	1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∅c0 4352   × cxp 5698  dom cdm 5700  (class class class)co 7448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-dm 5710  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  distrpi  10967  distrnq  11030  distrpr  11097  distrsr  11160