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Theorem eulerpartlemo 30752
Description: Lemma for eulerpart 30769: 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 5497 . . . 4 (𝑔 = 𝐴𝑔 = 𝐴)
21imaeq1d 5675 . . 3 (𝑔 = 𝐴 → (𝑔 “ ℕ) = (𝐴 “ ℕ))
32raleqdv 3333 . 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 3562 1 (𝐴𝑂 ↔ (𝐴𝑃 ∧ ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  {crab 3100   class class class wbr 4844  ccnv 5310  cima 5314  cfv 6101  (class class class)co 6874  𝑚 cmap 8092  Fincfn 8192  1c1 10222   · cmul 10226  cle 10360  cn 11305  2c2 11356  0cn0 11559  Σcsu 14639  cdvds 15203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324
This theorem is referenced by:  eulerpartlemr  30761
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