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Theorem oddpwdcv 30545
 Description: Lemma for eulerpart 30572: 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 7249 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
21fveq2d 6233 . 2 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩))
3 df-ov 6693 . . 3 ((1st𝑊)𝐹(2nd𝑊)) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩)
43a1i 11 . 2 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊)𝐹(2nd𝑊)) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩))
5 elxp6 7244 . . . 4 (𝑊 ∈ (𝐽 × ℕ0) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0)))
65simprbi 479 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0))
7 oveq2 6698 . . . 4 (𝑥 = (1st𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st𝑊)))
8 oveq2 6698 . . . . 5 (𝑦 = (2nd𝑊) → (2↑𝑦) = (2↑(2nd𝑊)))
98oveq1d 6705 . . . 4 (𝑦 = (2nd𝑊) → ((2↑𝑦) · (1st𝑊)) = ((2↑(2nd𝑊)) · (1st𝑊)))
10 oddpwdc.f . . . 4 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
11 ovex 6718 . . . 4 ((2↑(2nd𝑊)) · (1st𝑊)) ∈ V
127, 9, 10, 11ovmpt2 6838 . . 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 2691 1 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = ((2↑(2nd𝑊)) · (1st𝑊)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {crab 2945  ⟨cop 4216   class class class wbr 4685   × cxp 5141  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  1st c1st 7208  2nd c2nd 7209   · cmul 9979  ℕcn 11058  2c2 11108  ℕ0cn0 11330  ↑cexp 12900   ∥ cdvds 15027 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211 This theorem is referenced by:  eulerpartlemgvv  30566  eulerpartlemgh  30568  eulerpartlemgs2  30570
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