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Theorem ndmaovass 47156
Description: Any operation is associative outside its domain. In contrast to ndmovass 7621 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 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
21eleq2i 2831 . . . . 5 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5725 . . . . 5 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
42, 3bitri 275 . . . 4 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
5 aovvdm 47135 . . . . . 6 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
61eleq2i 2831 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
7 opelxp 5725 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
86, 7bitri 275 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴𝑆𝐵𝑆))
9 df-3an 1088 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
109simplbi2 500 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
118, 10sylbi 217 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
125, 11syl 17 . . . . 5 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1312imp 406 . . . 4 (( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
144, 13sylbi 217 . . 3 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
15 ndmaov 47133 . . 3 (¬ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
1614, 15nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
171eleq2i 2831 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
18 opelxp 5725 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
1917, 18bitri 275 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
20 aovvdm 47135 . . . . . . 7 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
211eleq2i 2831 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
22 opelxp 5725 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
2321, 22bitri 275 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
24 3anass 1094 . . . . . . . . . . 11 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
2524biimpri 228 . . . . . . . . . 10 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
2625a1d 25 . . . . . . . . 9 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
2726expcom 413 . . . . . . . 8 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
2823, 27sylbi 217 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
2920, 28syl 17 . . . . . 6 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
3029impcom 407 . . . . 5 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3119, 30sylbi 217 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3231pm2.43i 52 . . 3 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
33 ndmaov 47133 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3432, 33nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3516, 34eqtr4d 2778 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   × cxp 5687  dom cdm 5689   ((caov 47068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070  df-aov 47071
This theorem is referenced by: (None)
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