Theorem ndmaovdistr 44699

Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 7461 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	. . . . . 6 ⊢ dom 𝐺 = (𝑆 × 𝑆)
2	1	eleq2i 2830	. . . . 5 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
3		opelxp 5625	. . . . 5 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
4	2, 3	bitri 274	. . . 4 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ (𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
5		aovvdm 44677	. . . . . 6 ⊢ ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
6		ndmaov.1	. . . . . . . . 9 ⊢ dom 𝐹 = (𝑆 × 𝑆)
7	6	eleq2i 2830	. . . . . . . 8 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
8		opelxp 5625	. . . . . . . 8 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
9	7, 8	bitri 274	. . . . . . 7 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
10		3anass 1094	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
11	10	simplbi2com 503	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
12	9, 11	sylbi 216	. . . . . 6 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
13	5, 12	syl 17	. . . . 5 ⊢ ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
14	13	impcom 408	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
15	4, 14	sylbi 216	. . 3 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
16		ndmaov 44675	. . 3 ⊢ (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
17	15, 16	nsyl5 159	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
18	6	eleq2i 2830	. . . . 5 ⊢ (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆))
19		opelxp 5625	. . . . 5 ⊢ (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
20	18, 19	bitri 274	. . . 4 ⊢ (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
21		aovvdm 44677	. . . . . 6 ⊢ ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐺)
22	1	eleq2i 2830	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
23		opelxp 5625	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
24	22, 23	bitri 274	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
25	1	eleq2i 2830	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆))
26		opelxp 5625	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
27	25, 26	bitri 274	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ (𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
28		simpll 764	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ∈ 𝑆)
29		simprr 770	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐵 ∈ 𝑆)
30		simplr 766	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐶 ∈ 𝑆)
31	28, 29, 30	3jca 1127	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
32	31	ex 413	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
33	27, 32	sylbi 216	. . . . . . . 8 ⊢ (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
34		aovvdm 44677	. . . . . . . 8 ⊢ ( ((𝐴𝐺𝐶)) ∈ 𝑆 → ⟨𝐴, 𝐶⟩ ∈ dom 𝐺)
35	33, 34	syl11 33	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
36	24, 35	sylbi 216	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
37	21, 36	syl 17	. . . . 5 ⊢ ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
38	37	imp 407	. . . 4 ⊢ (( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
39	20, 38	sylbi 216	. . 3 ⊢ (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
40		ndmaov 44675	. . 3 ⊢ (¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
41	39, 40	nsyl5 159	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
42	17, 41	eqtr4d 2781	1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   × cxp 5587  dom cdm 5589   ((caov 44610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-aiota 44577  df-dfat 44611  df-afv 44612  df-aov 44613

This theorem is referenced by: (None)