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Theorem ndmaovdistr 43847
 Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 7325 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 2881 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5558 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
42, 3bitri 278 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
5 aovvdm 43825 . . . . . 6 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
6 ndmaov.1 . . . . . . . . 9 dom 𝐹 = (𝑆 × 𝑆)
76eleq2i 2881 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
8 opelxp 5558 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
97, 8bitri 278 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
10 3anass 1092 . . . . . . . 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 43823 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
1715, 16nsyl5 162 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
186eleq2i 2881 . . . . 5 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆))
19 opelxp 5558 . . . . 5 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
2018, 19bitri 278 . . . 4 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
21 aovvdm 43825 . . . . . 6 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐺)
221eleq2i 2881 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
23 opelxp 5558 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2422, 23bitri 278 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐵𝑆))
251eleq2i 2881 . . . . . . . . . 10 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆))
26 opelxp 5558 . . . . . . . . . 10 (⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐶𝑆))
2725, 26bitri 278 . . . . . . . . 9 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐶𝑆))
28 simpll 766 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
29 simprr 772 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
30 simplr 768 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐶𝑆)
3128, 29, 303jca 1125 . . . . . . . . . 10 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
3231ex 416 . . . . . . . . 9 ((𝐴𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3327, 32sylbi 220 . . . . . . . 8 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
34 aovvdm 43825 . . . . . . . 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 43823 . . 3 (¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4139, 40nsyl5 162 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4217, 41eqtr4d 2836 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⟨cop 4533   × cxp 5520  dom cdm 5522   ((caov 43758 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-res 5534  df-iota 6288  df-fun 6331  df-fv 6337  df-aiota 43726  df-dfat 43759  df-afv 43760  df-aov 43761 This theorem is referenced by: (None)
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