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Theorem ndmaovass 43692
Description: Any operation is associative outside its domain. In contrast to ndmovass 7330 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 2907 . . . . 5 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5578 . . . . 5 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
42, 3bitri 278 . . . 4 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
5 aovvdm 43671 . . . . . 6 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
61eleq2i 2907 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
7 opelxp 5578 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
86, 7bitri 278 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴𝑆𝐵𝑆))
9 df-3an 1086 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
109simplbi2 504 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
118, 10sylbi 220 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
125, 11syl 17 . . . . 5 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1312imp 410 . . . 4 (( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
144, 13sylbi 220 . . 3 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
15 ndmaov 43669 . . 3 (¬ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
1614, 15nsyl5 162 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
171eleq2i 2907 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
18 opelxp 5578 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
1917, 18bitri 278 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
20 aovvdm 43671 . . . . . . 7 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
211eleq2i 2907 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
22 opelxp 5578 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
2321, 22bitri 278 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
24 3anass 1092 . . . . . . . . . . 11 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
2524biimpri 231 . . . . . . . . . 10 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
2625a1d 25 . . . . . . . . 9 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
2726expcom 417 . . . . . . . 8 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
2823, 27sylbi 220 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
2920, 28syl 17 . . . . . 6 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
3029impcom 411 . . . . 5 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3119, 30sylbi 220 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3231pm2.43i 52 . . 3 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
33 ndmaov 43669 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3432, 33nsyl5 162 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3516, 34eqtr4d 2862 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3480  cop 4556   × cxp 5540  dom cdm 5542   ((caov 43604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-aiota 43572  df-dfat 43605  df-afv 43606  df-aov 43607
This theorem is referenced by: (None)
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