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Theorem ndmaovdistr 44371
Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 7397 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 . . . . . 6 dom 𝐺 = (𝑆 × 𝑆)
21eleq2i 2829 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5587 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
42, 3bitri 278 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
5 aovvdm 44349 . . . . . 6 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
6 ndmaov.1 . . . . . . . . 9 dom 𝐹 = (𝑆 × 𝑆)
76eleq2i 2829 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
8 opelxp 5587 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
97, 8bitri 278 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
10 3anass 1097 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
1110simplbi2com 506 . . . . . . 7 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
129, 11sylbi 220 . . . . . 6 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
135, 12syl 17 . . . . 5 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1413impcom 411 . . . 4 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
154, 14sylbi 220 . . 3 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → (𝐴𝑆𝐵𝑆𝐶𝑆))
16 ndmaov 44347 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
1715, 16nsyl5 162 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
186eleq2i 2829 . . . . 5 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆))
19 opelxp 5587 . . . . 5 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
2018, 19bitri 278 . . . 4 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
21 aovvdm 44349 . . . . . 6 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐺)
221eleq2i 2829 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
23 opelxp 5587 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2422, 23bitri 278 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐵𝑆))
251eleq2i 2829 . . . . . . . . . 10 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆))
26 opelxp 5587 . . . . . . . . . 10 (⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐶𝑆))
2725, 26bitri 278 . . . . . . . . 9 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐶𝑆))
28 simpll 767 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
29 simprr 773 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
30 simplr 769 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐶𝑆)
3128, 29, 303jca 1130 . . . . . . . . . 10 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
3231ex 416 . . . . . . . . 9 ((𝐴𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3327, 32sylbi 220 . . . . . . . 8 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
34 aovvdm 44349 . . . . . . . 8 ( ((𝐴𝐺𝐶)) ∈ 𝑆 → ⟨𝐴, 𝐶⟩ ∈ dom 𝐺)
3533, 34syl11 33 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3624, 35sylbi 220 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3721, 36syl 17 . . . . 5 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3837imp 410 . . . 4 (( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
3920, 38sylbi 220 . . 3 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
40 ndmaov 44347 . . 3 (¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4139, 40nsyl5 162 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4217, 41eqtr4d 2780 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  Vcvv 3408  cop 4547   × cxp 5549  dom cdm 5551   ((caov 44282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-res 5563  df-iota 6338  df-fun 6382  df-fv 6388  df-aiota 44249  df-dfat 44283  df-afv 44284  df-aov 44285
This theorem is referenced by: (None)
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