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Theorem eulerpartlemo 33195
Description: Lemma for eulerpart 33212: 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 5865 . . . 4 (𝑔 = 𝐴𝑔 = 𝐴)
21imaeq1d 6048 . . 3 (𝑔 = 𝐴 → (𝑔 “ ℕ) = (𝐴 “ ℕ))
32raleqdv 3324 . 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 3682 1 (𝐴𝑂 ↔ (𝐴𝑃 ∧ ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  {crab 3431   class class class wbr 5141  ccnv 5668  cima 5672  cfv 6532  (class class class)co 7393  m cmap 8803  Fincfn 8922  1c1 11093   · cmul 11097  cle 11231  cn 12194  2c2 12249  0cn0 12454  Σcsu 15614  cdvds 16179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682
This theorem is referenced by:  eulerpartlemr  33204
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