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Theorem eulerpartlemo 30763
Description: Lemma for eulerpart 30780: 𝑂 is the set of odd partitions of 𝑁. (Contributed by Thierry Arnoux, 10-Aug-2017.)
Hypotheses
Ref Expression
eulerpart.p 𝑃 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
eulerpart.o 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
Assertion
Ref Expression
eulerpartlemo (𝐴𝑂 ↔ (𝐴𝑃 ∧ ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
Distinct variable groups:   𝑔,𝑛,𝐴   𝑃,𝑔
Allowed substitution hints:   𝐴(𝑓,𝑘)   𝐷(𝑓,𝑔,𝑘,𝑛)   𝑃(𝑓,𝑘,𝑛)   𝑁(𝑓,𝑔,𝑘,𝑛)   𝑂(𝑓,𝑔,𝑘,𝑛)

Proof of Theorem eulerpartlemo
StepHypRef Expression
1 cnveq 5432 . . . 4 (𝑔 = 𝐴𝑔 = 𝐴)
21imaeq1d 5604 . . 3 (𝑔 = 𝐴 → (𝑔 “ ℕ) = (𝐴 “ ℕ))
32raleqdv 3293 . 2 (𝑔 = 𝐴 → (∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
4 eulerpart.o . 2 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
53, 4elrab2 3518 1 (𝐴𝑂 ↔ (𝐴𝑃 ∧ ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065   class class class wbr 4786  ccnv 5248  cima 5252  cfv 6029  (class class class)co 6792  𝑚 cmap 8009  Fincfn 8109  1c1 10139   · cmul 10143  cle 10277  cn 11222  2c2 11272  0cn0 11495  Σcsu 14620  cdvds 15185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262
This theorem is referenced by:  eulerpartlemr  30772
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