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Theorem ndmaovdistr 46646
Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 7604 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 2817 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆))
3 opelxp 5709 . . . . 5 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
42, 3bitri 274 . . . 4 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 ↔ (𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆))
5 aovvdm 46624 . . . . . 6 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → ⟨𝐵, 𝐶⟩ ∈ dom 𝐹)
6 ndmaov.1 . . . . . . . . 9 dom 𝐹 = (𝑆 × 𝑆)
76eleq2i 2817 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆))
8 opelxp 5709 . . . . . . . 8 (⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐶𝑆))
97, 8bitri 274 . . . . . . 7 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 ↔ (𝐵𝑆𝐶𝑆))
10 3anass 1092 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
1110simplbi2com 501 . . . . . . 7 ((𝐵𝑆𝐶𝑆) → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
129, 11sylbi 216 . . . . . 6 (⟨𝐵, 𝐶⟩ ∈ dom 𝐹 → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
135, 12syl 17 . . . . 5 ( ((𝐵𝐹𝐶)) ∈ 𝑆 → (𝐴𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
1413impcom 406 . . . 4 ((𝐴𝑆 ∧ ((𝐵𝐹𝐶)) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
154, 14sylbi 216 . . 3 (⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → (𝐴𝑆𝐵𝑆𝐶𝑆))
16 ndmaov 46622 . . 3 (¬ ⟨𝐴, ((𝐵𝐹𝐶)) ⟩ ∈ dom 𝐺 → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
1715, 16nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = V)
186eleq2i 2817 . . . . 5 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆))
19 opelxp 5709 . . . . 5 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ (𝑆 × 𝑆) ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
2018, 19bitri 274 . . . 4 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 ↔ ( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆))
21 aovvdm 46624 . . . . . 6 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐺)
221eleq2i 2817 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
23 opelxp 5709 . . . . . . . 8 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2422, 23bitri 274 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐵𝑆))
251eleq2i 2817 . . . . . . . . . 10 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ ⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆))
26 opelxp 5709 . . . . . . . . . 10 (⟨𝐴, 𝐶⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐶𝑆))
2725, 26bitri 274 . . . . . . . . 9 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 ↔ (𝐴𝑆𝐶𝑆))
28 simpll 765 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
29 simprr 771 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
30 simplr 767 . . . . . . . . . . 11 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → 𝐶𝑆)
3128, 29, 303jca 1125 . . . . . . . . . 10 (((𝐴𝑆𝐶𝑆) ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝑆𝐵𝑆𝐶𝑆))
3231ex 411 . . . . . . . . 9 ((𝐴𝑆𝐶𝑆) → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3327, 32sylbi 216 . . . . . . . 8 (⟨𝐴, 𝐶⟩ ∈ dom 𝐺 → ((𝐴𝑆𝐵𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆)))
34 aovvdm 46624 . . . . . . . 8 ( ((𝐴𝐺𝐶)) ∈ 𝑆 → ⟨𝐴, 𝐶⟩ ∈ dom 𝐺)
3533, 34syl11 33 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3624, 35sylbi 216 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ dom 𝐺 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3721, 36syl 17 . . . . 5 ( ((𝐴𝐺𝐵)) ∈ 𝑆 → ( ((𝐴𝐺𝐶)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆𝐶𝑆)))
3837imp 405 . . . 4 (( ((𝐴𝐺𝐵)) ∈ 𝑆 ∧ ((𝐴𝐺𝐶)) ∈ 𝑆) → (𝐴𝑆𝐵𝑆𝐶𝑆))
3920, 38sylbi 216 . . 3 (⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (𝐴𝑆𝐵𝑆𝐶𝑆))
40 ndmaov 46622 . . 3 (¬ ⟨ ((𝐴𝐺𝐵)) , ((𝐴𝐺𝐶)) ⟩ ∈ dom 𝐹 → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4139, 40nsyl5 159 . 2 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) = V)
4217, 41eqtr4d 2768 1 (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3463  cop 4631   × cxp 5671  dom cdm 5673   ((caov 46557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-res 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-aiota 46524  df-dfat 46558  df-afv 46559  df-aov 46560
This theorem is referenced by: (None)
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