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Theorem oddpwdcv 34357
Description: Lemma for eulerpart 34384: value of the 𝐹 function. (Contributed by Thierry Arnoux, 9-Sep-2017.)
Hypotheses
Ref Expression
oddpwdc.j 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
oddpwdc.f 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
Assertion
Ref Expression
oddpwdcv (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = ((2↑(2nd𝑊)) · (1st𝑊)))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝐽,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧)   𝐽(𝑧)   𝑊(𝑧)

Proof of Theorem oddpwdcv
StepHypRef Expression
1 1st2nd2 8053 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
21fveq2d 6910 . 2 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩))
3 df-ov 7434 . . 3 ((1st𝑊)𝐹(2nd𝑊)) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩)
43a1i 11 . 2 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊)𝐹(2nd𝑊)) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩))
5 elxp6 8048 . . . 4 (𝑊 ∈ (𝐽 × ℕ0) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0)))
65simprbi 496 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0))
7 oveq2 7439 . . . 4 (𝑥 = (1st𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st𝑊)))
8 oveq2 7439 . . . . 5 (𝑦 = (2nd𝑊) → (2↑𝑦) = (2↑(2nd𝑊)))
98oveq1d 7446 . . . 4 (𝑦 = (2nd𝑊) → ((2↑𝑦) · (1st𝑊)) = ((2↑(2nd𝑊)) · (1st𝑊)))
10 oddpwdc.f . . . 4 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
11 ovex 7464 . . . 4 ((2↑(2nd𝑊)) · (1st𝑊)) ∈ V
127, 9, 10, 11ovmpo 7593 . . 3 (((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0) → ((1st𝑊)𝐹(2nd𝑊)) = ((2↑(2nd𝑊)) · (1st𝑊)))
136, 12syl 17 . 2 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊)𝐹(2nd𝑊)) = ((2↑(2nd𝑊)) · (1st𝑊)))
142, 4, 133eqtr2d 2783 1 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = ((2↑(2nd𝑊)) · (1st𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  cop 4632   class class class wbr 5143   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013   · cmul 11160  cn 12266  2c2 12321  0cn0 12526  cexp 14102  cdvds 16290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015
This theorem is referenced by:  eulerpartlemgvv  34378  eulerpartlemgh  34380  eulerpartlemgs2  34382
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