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Theorem ndmaovdistr 41608
Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 6865 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.)
Hypotheses
Ref Expression
ndmaov.1 dom 𝐹 = (𝑆 × 𝑆)
ndmaov.6 dom 𝐺 = (𝑆 × 𝑆)
Assertion
Ref Expression
ndmaovdistr (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )

Proof of Theorem ndmaovdistr
StepHypRef Expression
1 ndmaov.6 . . . . . . 7 dom 𝐺 = (𝑆 × 𝑆)
21eleq2i 2722 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5180 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
42, 3bitri 264 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
5 aovvdm 41586 . . . . . . 7 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
6 ndmaov.1 . . . . . . . . . 10 dom 𝐹 = (𝑆 × 𝑆)
76eleq2i 2722 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
8 opelxp 5180 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
97, 8bitri 264 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
10 3anass 1059 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
1110simplbi2com 656 . . . . . . . 8 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
129, 11sylbi 207 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
135, 12syl 17 . . . . . 6 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1413impcom 445 . . . . 5 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
154, 14sylbi 207 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → (𝐴𝑆𝐵𝑆𝐶𝑆))
1615con3i 150 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺)
17 ndmaov 41584 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
1816, 17syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
196eleq2i 2722 . . . . . 6 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆))
20 opelxp 5180 . . . . . 6 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
2119, 20bitri 264 . . . . 5 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
22 aovvdm 41586 . . . . . . 7 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐺)
231eleq2i 2722 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
24 opelxp 5180 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2523, 24bitri 264 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐵𝑆))
261eleq2i 2722 . . . . . . . . . . 11 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆))
27 opelxp 5180 . . . . . . . . . . 11 (⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐶𝑆))
2826, 27bitri 264 . . . . . . . . . 10 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐶𝑆))
29 simpll 805 . . . . . . . . . . . 12 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
30 simprr 811 . . . . . . . . . . . 12 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
31 simplr 807 . . . . . . . . . . . 12 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐶𝑆)
3229, 30, 313jca 1261 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
3332ex 449 . . . . . . . . . 10 ((𝐴𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3428, 33sylbi 207 . . . . . . . . 9 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
35 aovvdm 41586 . . . . . . . . 9 ( ((𝐴𝐺𝐶)) ∈ 𝑆 → ⟨𝐴, 𝐶⟩ ∈ dom 𝐺)
3634, 35syl11 33 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3725, 36sylbi 207 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3822, 37syl 17 . . . . . 6 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3938imp 444 . . . . 5 (( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
4021, 39sylbi 207 . . . 4 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
4140con3i 150 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹)
42 ndmaov 41584 . . 3 (¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4341, 42syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4418, 43eqtr4d 2688 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216   × cxp 5141  dom cdm 5143   ((caov 41516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-fv 5934  df-dfat 41517  df-afv 41518  df-aov 41519
This theorem is referenced by: (None)
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