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Theorem ndmaovass 39935
 Description: Any operation is associative outside its domain. In contrast to ndmovass 6720 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 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )

Proof of Theorem ndmaovass
StepHypRef Expression
1 ndmaov.1 . . . . . . 7 dom 𝐹 = (𝑆 × 𝑆)
21eleq2i 2680 . . . . . 6 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5070 . . . . . 6 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
42, 3bitri 263 . . . . 5 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
5 aovvdm 39914 . . . . . . 7 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
61eleq2i 2680 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
7 opelxp 5070 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
86, 7bitri 263 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴𝑆𝐵𝑆))
9 df-3an 1033 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
109simplbi2 653 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
118, 10sylbi 206 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
125, 11syl 17 . . . . . 6 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1312imp 444 . . . . 5 (( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
144, 13sylbi 206 . . . 4 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
1514con3i 149 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹)
16 ndmaov 39912 . . 3 (¬ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
1715, 16syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
181eleq2i 2680 . . . . . . 7 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
19 opelxp 5070 . . . . . . 7 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
2018, 19bitri 263 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
21 aovvdm 39914 . . . . . . . 8 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
221eleq2i 2680 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
23 opelxp 5070 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
2422, 23bitri 263 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
25 3anass 1035 . . . . . . . . . . . 12 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
2625biimpri 217 . . . . . . . . . . 11 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
2726a1d 25 . . . . . . . . . 10 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
2827expcom 450 . . . . . . . . 9 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
2924, 28sylbi 206 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
3021, 29syl 17 . . . . . . 7 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
3130impcom 445 . . . . . 6 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3220, 31sylbi 206 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3332pm2.43i 50 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
3433con3i 149 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹)
35 ndmaov 39912 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3634, 35syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3717, 36eqtr4d 2647 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   × cxp 5036  dom cdm 5038   ((caov 39844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-fv 5812  df-dfat 39845  df-afv 39846  df-aov 39847 This theorem is referenced by: (None)
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