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Theorem ndmaovass 41049
Description: Any operation is associative outside its domain. In contrast to ndmovass 6807 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 2691 . . . . . 6 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5136 . . . . . 6 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
42, 3bitri 264 . . . . 5 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
5 aovvdm 41028 . . . . . . 7 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
61eleq2i 2691 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
7 opelxp 5136 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
86, 7bitri 264 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴𝑆𝐵𝑆))
9 df-3an 1038 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
109simplbi2 654 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
118, 10sylbi 207 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
125, 11syl 17 . . . . . 6 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1312imp 445 . . . . 5 (( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
144, 13sylbi 207 . . . 4 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
1514con3i 150 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹)
16 ndmaov 41026 . . 3 (¬ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
1715, 16syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
181eleq2i 2691 . . . . . . 7 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
19 opelxp 5136 . . . . . . 7 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
2018, 19bitri 264 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
21 aovvdm 41028 . . . . . . . 8 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
221eleq2i 2691 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
23 opelxp 5136 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
2422, 23bitri 264 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
25 3anass 1040 . . . . . . . . . . . 12 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
2625biimpri 218 . . . . . . . . . . 11 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
2726a1d 25 . . . . . . . . . 10 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
2827expcom 451 . . . . . . . . 9 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
2924, 28sylbi 207 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
3021, 29syl 17 . . . . . . 7 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
3130impcom 446 . . . . . 6 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3220, 31sylbi 207 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3332pm2.43i 52 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
3433con3i 150 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹)
35 ndmaov 41026 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3634, 35syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3717, 36eqtr4d 2657 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174   × cxp 5102  dom cdm 5104   ((caov 40958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-xp 5110  df-fv 5884  df-dfat 40959  df-afv 40960  df-aov 40961
This theorem is referenced by: (None)
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