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Theorem oddpwdcv 34337
Description: Lemma for eulerpart 34364: 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 8052 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
21fveq2d 6911 . 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 8047 . . . 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 2781 1 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = ((2↑(2nd𝑊)) · (1st𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  cop 4637   class class class wbr 5148   × cxp 5687  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012   · cmul 11158  cn 12264  2c2 12319  0cn0 12524  cexp 14099  cdvds 16287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014
This theorem is referenced by:  eulerpartlemgvv  34358  eulerpartlemgh  34360  eulerpartlemgs2  34362
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