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Theorem ndmaovdistr 42252
 Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 7100 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 2851 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5391 . . . . . 6 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
42, 3bitri 267 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
5 aovvdm 42230 . . . . . . 7 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
6 ndmaov.1 . . . . . . . . . 10 dom 𝐹 = (𝑆 × 𝑆)
76eleq2i 2851 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
8 opelxp 5391 . . . . . . . . 9 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
97, 8bitri 267 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
10 3anass 1079 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
1110simplbi2com 498 . . . . . . . 8 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
129, 11sylbi 209 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
135, 12syl 17 . . . . . 6 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1413impcom 398 . . . . 5 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
154, 14sylbi 209 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → (𝐴𝑆𝐵𝑆𝐶𝑆))
1615con3i 152 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺)
17 ndmaov 42228 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
1816, 17syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
196eleq2i 2851 . . . . . 6 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆))
20 opelxp 5391 . . . . . 6 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
2119, 20bitri 267 . . . . 5 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
22 aovvdm 42230 . . . . . . 7 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐺)
231eleq2i 2851 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
24 opelxp 5391 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2523, 24bitri 267 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐵𝑆))
261eleq2i 2851 . . . . . . . . . . 11 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆))
27 opelxp 5391 . . . . . . . . . . 11 (⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐶𝑆))
2826, 27bitri 267 . . . . . . . . . 10 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐶𝑆))
29 simpll 757 . . . . . . . . . . . 12 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
30 simprr 763 . . . . . . . . . . . 12 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
31 simplr 759 . . . . . . . . . . . 12 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐶𝑆)
3229, 30, 313jca 1119 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
3332ex 403 . . . . . . . . . 10 ((𝐴𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3428, 33sylbi 209 . . . . . . . . 9 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
35 aovvdm 42230 . . . . . . . . 9 ( ((𝐴𝐺𝐶)) ∈ 𝑆 → ⟨𝐴, 𝐶⟩ ∈ dom 𝐺)
3634, 35syl11 33 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3725, 36sylbi 209 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3822, 37syl 17 . . . . . 6 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3938imp 397 . . . . 5 (( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
4021, 39sylbi 209 . . . 4 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
4140con3i 152 . . 3 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹)
42 ndmaov 42228 . . 3 (¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4341, 42syl 17 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4418, 43eqtr4d 2817 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ⟨cop 4404   × cxp 5353  dom cdm 5355   ((caov 42163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-res 5367  df-iota 6099  df-fun 6137  df-fv 6143  df-aiota 42119  df-dfat 42164  df-afv 42165  df-aov 42166 This theorem is referenced by: (None)
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