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Theorem oddpwdcv 31186
 Description: Lemma for eulerpart 31213: 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 7575 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
21fveq2d 6534 . 2 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩))
3 df-ov 7010 . . 3 ((1st𝑊)𝐹(2nd𝑊)) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩)
43a1i 11 . 2 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊)𝐹(2nd𝑊)) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩))
5 elxp6 7570 . . . 4 (𝑊 ∈ (𝐽 × ℕ0) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0)))
65simprbi 497 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0))
7 oveq2 7015 . . . 4 (𝑥 = (1st𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st𝑊)))
8 oveq2 7015 . . . . 5 (𝑦 = (2nd𝑊) → (2↑𝑦) = (2↑(2nd𝑊)))
98oveq1d 7022 . . . 4 (𝑦 = (2nd𝑊) → ((2↑𝑦) · (1st𝑊)) = ((2↑(2nd𝑊)) · (1st𝑊)))
10 oddpwdc.f . . . 4 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
11 ovex 7039 . . . 4 ((2↑(2nd𝑊)) · (1st𝑊)) ∈ V
127, 9, 10, 11ovmpo 7157 . . 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 2835 1 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = ((2↑(2nd𝑊)) · (1st𝑊)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1520   ∈ wcel 2079  {crab 3107  ⟨cop 4472   class class class wbr 4956   × cxp 5433  ‘cfv 6217  (class class class)co 7007   ∈ cmpo 7009  1st c1st 7534  2nd c2nd 7535   · cmul 10377  ℕcn 11475  2c2 11529  ℕ0cn0 11734  ↑cexp 13267   ∥ cdvds 15428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-iota 6181  df-fun 6219  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-1st 7536  df-2nd 7537 This theorem is referenced by:  eulerpartlemgvv  31207  eulerpartlemgh  31209  eulerpartlemgs2  31211
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