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Theorem oddpwdcv 29547
 Description: Lemma for eulerpart 29574: 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 7070 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
21fveq2d 6089 . 2 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩))
3 df-ov 6527 . . 3 ((1st𝑊)𝐹(2nd𝑊)) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩)
43a1i 11 . 2 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊)𝐹(2nd𝑊)) = (𝐹‘⟨(1st𝑊), (2nd𝑊)⟩))
5 elxp6 7065 . . . 4 (𝑊 ∈ (𝐽 × ℕ0) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0)))
65simprbi 478 . . 3 (𝑊 ∈ (𝐽 × ℕ0) → ((1st𝑊) ∈ 𝐽 ∧ (2nd𝑊) ∈ ℕ0))
7 oveq2 6532 . . . 4 (𝑥 = (1st𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st𝑊)))
8 oveq2 6532 . . . . 5 (𝑦 = (2nd𝑊) → (2↑𝑦) = (2↑(2nd𝑊)))
98oveq1d 6539 . . . 4 (𝑦 = (2nd𝑊) → ((2↑𝑦) · (1st𝑊)) = ((2↑(2nd𝑊)) · (1st𝑊)))
10 oddpwdc.f . . . 4 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
11 ovex 6552 . . . 4 ((2↑(2nd𝑊)) · (1st𝑊)) ∈ V
127, 9, 10, 11ovmpt2 6669 . . 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 2646 1 (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = ((2↑(2nd𝑊)) · (1st𝑊)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   = wceq 1474   ∈ wcel 1976  {crab 2896  ⟨cop 4127   class class class wbr 4574   × cxp 5023  ‘cfv 5787  (class class class)co 6524   ↦ cmpt2 6526  1st c1st 7031  2nd c2nd 7032   · cmul 9794  ℕcn 10864  2c2 10914  ℕ0cn0 11136  ↑cexp 12674   ∥ cdvds 14764 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-iota 5751  df-fun 5789  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034 This theorem is referenced by:  eulerpartlemgvv  29568  eulerpartlemgh  29570  eulerpartlemgs2  29572
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