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Theorem ndmaovass 44968
Description: Any operation is associative outside its domain. In contrast to ndmovass 7501 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 2828 . . . . 5 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5643 . . . . 5 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
42, 3bitri 274 . . . 4 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆))
5 aovvdm 44947 . . . . . 6 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
61eleq2i 2828 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
7 opelxp 5643 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
86, 7bitri 274 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴𝑆𝐵𝑆))
9 df-3an 1088 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
109simplbi2 501 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
118, 10sylbi 216 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
125, 11syl 17 . . . . 5 ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐶𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1312imp 407 . . . 4 (( ((𝐴𝐹𝐵)) ∈ 𝑆𝐶𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
144, 13sylbi 216 . . 3 (⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
15 ndmaov 44945 . . 3 (¬ ⟨ ((𝐴𝐹𝐵)) , 𝐶⟩ ∈ dom 𝐹 → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
1614, 15nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = V)
171eleq2i 2828 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
18 opelxp 5643 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
1917, 18bitri 274 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
20 aovvdm 44947 . . . . . . 7 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
211eleq2i 2828 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
22 opelxp 5643 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
2321, 22bitri 274 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
24 3anass 1094 . . . . . . . . . . 11 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
2524biimpri 227 . . . . . . . . . 10 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
2625a1d 25 . . . . . . . . 9 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
2726expcom 414 . . . . . . . 8 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
2823, 27sylbi 216 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
2920, 28syl 17 . . . . . 6 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))))
3029impcom 408 . . . . 5 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3119, 30sylbi 216 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3231pm2.43i 52 . . 3 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
33 ndmaov 44945 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐹 → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3432, 33nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐹 ((𝐵𝐹𝐶)) )) = V)
3516, 34eqtr4d 2779 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105  Vcvv 3440  cop 4576   × cxp 5605  dom cdm 5607   ((caov 44880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-int 4892  df-br 5087  df-opab 5149  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-res 5619  df-iota 6417  df-fun 6467  df-fv 6473  df-aiota 44847  df-dfat 44881  df-afv 44882  df-aov 44883
This theorem is referenced by: (None)
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