Theorem ndmaovass 47215

Description: Any operation is associative outside its domain. In contrast to ndmovass 7600 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.)

Hypothesis

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

Assertion

Ref	Expression
ndmaovass	⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )

Proof of Theorem ndmaovass

Step	Hyp	Ref	Expression
1		ndmaov.1	. . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆)
2	1	eleq2i 2827	. . . . 5 ⊢ (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆))
3		opelxp 5695	. . . . 5 ⊢ (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
4	2, 3	bitri 275	. . . 4 ⊢ (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
5		aovvdm 47194	. . . . . 6 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6	1	eleq2i 2827	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
7		opelxp 5695	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
8	6, 7	bitri 275	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
9		df-3an 1088	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆))
10	9	simplbi2 500	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
11	8, 10	sylbi 217	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐶 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
12	5, 11	syl 17	. . . . 5 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐶 ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
13	12	imp 406	. . . 4 ⊢ (( ((𝐴𝐹𝐵)) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
14	4, 13	sylbi 217	. . 3 ⊢ (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
15		ndmaov 47192	. . 3 ⊢ (¬ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
16	14, 15	nsyl5 159	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
17	1	eleq2i 2827	. . . . . 6 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
18		opelxp 5695	. . . . . 6 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
19	17, 18	bitri 275	. . . . 5 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ (𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
20		aovvdm 47194	. . . . . . 7 ⊢ ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
21	1	eleq2i 2827	. . . . . . . . 9 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
22		opelxp 5695	. . . . . . . . 9 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
23	21, 22	bitri 275	. . . . . . . 8 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
24		3anass 1094	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
25	24	biimpri 228	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
26	25	a1d 25	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
27	26	expcom 413	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ∈ 𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))))
28	23, 27	sylbi 217	. . . . . . 7 ⊢ (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))))
29	20, 28	syl 17	. . . . . 6 ⊢ ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴 ∈ 𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))))
30	29	impcom 407	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
31	19, 30	sylbi 217	. . . 4 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
32	31	pm2.43i 52	. . 3 ⊢ (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
33		ndmaov 47192	. . 3 ⊢ (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
34	32, 33	nsyl5 159	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
35	16, 34	eqtr4d 2774	1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ⟨cop 4612   × cxp 5657  dom cdm 5659   ((caov 47127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6489  df-fun 6538  df-fv 6544  df-aiota 47094  df-dfat 47128  df-afv 47129  df-aov 47130

This theorem is referenced by: (None)