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Theorem eulerpartlemo 29547
Description: Lemma for eulerpart 29564: 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 5205 . . . 4 (𝑔 = 𝐴𝑔 = 𝐴)
21imaeq1d 5370 . . 3 (𝑔 = 𝐴 → (𝑔 “ ℕ) = (𝐴 “ ℕ))
32raleqdv 3120 . 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 3332 1 (𝐴𝑂 ↔ (𝐴𝑃 ∧ ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  {crab 2899   class class class wbr 4577  ccnv 5026  cima 5030  cfv 5789  (class class class)co 6526  𝑚 cmap 7721  Fincfn 7818  1c1 9793   · cmul 9797  cle 9931  cn 10869  2c2 10919  0cn0 11141  Σcsu 14212  cdvds 14769
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-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
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-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040
This theorem is referenced by:  eulerpartlemr  29556
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