Theorem ndmaovdistr 42252

Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 7100 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
ndmaov.1	⊢ dom 𝐹 = (𝑆 × 𝑆)
ndmaov.6	⊢ dom 𝐺 = (𝑆 × 𝑆)

Assertion

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

Proof of Theorem ndmaovdistr

Step	Hyp	Ref	Expression
1		ndmaov.6	. . . . . . 7 ⊢ dom 𝐺 = (𝑆 × 𝑆)
2	1	eleq2i 2851	. . . . . 6 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
3		opelxp 5391	. . . . . 6 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
4	2, 3	bitri 267	. . . . 5 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ (𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
5		aovvdm 42230	. . . . . . 7 ⊢ ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
6		ndmaov.1	. . . . . . . . . 10 ⊢ dom 𝐹 = (𝑆 × 𝑆)
7	6	eleq2i 2851	. . . . . . . . 9 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
8		opelxp 5391	. . . . . . . . 9 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
9	7, 8	bitri 267	. . . . . . . 8 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
10		3anass 1079	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
11	10	simplbi2com 498	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
12	9, 11	sylbi 209	. . . . . . 7 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
13	5, 12	syl 17	. . . . . 6 ⊢ ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
14	13	impcom 398	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
15	4, 14	sylbi 209	. . . 4 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
16	15	con3i 152	. . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺)
17		ndmaov 42228	. . 3 ⊢ (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
18	16, 17	syl 17	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
19	6	eleq2i 2851	. . . . . 6 ⊢ (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆))
20		opelxp 5391	. . . . . 6 ⊢ (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
21	19, 20	bitri 267	. . . . 5 ⊢ (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
22		aovvdm 42230	. . . . . . 7 ⊢ ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐺)
23	1	eleq2i 2851	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
24		opelxp 5391	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
25	23, 24	bitri 267	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
26	1	eleq2i 2851	. . . . . . . . . . 11 ⊢ (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆))
27		opelxp 5391	. . . . . . . . . . 11 ⊢ (⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
28	26, 27	bitri 267	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
29		simpll 757	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ∈ 𝑆)
30		simprr 763	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐵 ∈ 𝑆)
31		simplr 759	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐶 ∈ 𝑆)
32	29, 30, 31	3jca 1119	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
33	32	ex 403	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
34	28, 33	sylbi 209	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
35		aovvdm 42230	. . . . . . . . 9 ⊢ ( ((𝐴𝐺𝐶)) ∈ 𝑆 → ⟨𝐴, 𝐶⟩ ∈ dom 𝐺)
36	34, 35	syl11 33	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
37	25, 36	sylbi 209	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
38	22, 37	syl 17	. . . . . 6 ⊢ ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
39	38	imp 397	. . . . 5 ⊢ (( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
40	21, 39	sylbi 209	. . . 4 ⊢ (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
41	40	con3i 152	. . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹)
42		ndmaov 42228	. . 3 ⊢ (¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
43	41, 42	syl 17	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
44	18, 43	eqtr4d 2817	1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ⟨cop 4404   × cxp 5353  dom cdm 5355   ((caov 42163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-res 5367  df-iota 6099  df-fun 6137  df-fv 6143  df-aiota 42119  df-dfat 42164  df-afv 42165  df-aov 42166

This theorem is referenced by: (None)