Theorem eulerpartlemo 31618

Description: Lemma for eulerpart 31635: 𝑂 is the set of odd partitions of 𝑁. (Contributed by Thierry Arnoux, 10-Aug-2017.)

Hypotheses

Ref	Expression
eulerpart.p	⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
eulerpart.o	⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d	⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}

Assertion

Ref	Expression
eulerpartlemo	⊢ (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ ∀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

Step	Hyp	Ref	Expression
1		cnveq 5738	. . . 4 ⊢ (𝑔 = 𝐴 → ◡𝑔 = ◡𝐴)
2	1	imaeq1d 5922	. . 3 ⊢ (𝑔 = 𝐴 → (◡𝑔 “ ℕ) = (◡𝐴 “ ℕ))
3	2	raleqdv 3415	. 2 ⊢ (𝑔 = 𝐴 → (∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
4		eulerpart.o	. 2 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
5	3, 4	elrab2 3682	1 ⊢ (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142   class class class wbr 5058  ^◡ccnv 5548   “ cima 5552  ‘cfv 6349  (class class class)co 7150   ↑_m cmap 8400  Fincfn 8503  1c1 10532   · cmul 10536   ≤ cle 10670  ℕcn 11632  2c2 11686  ℕ₀cn0 11891  Σcsu 15036   ∥ cdvds 15601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562

This theorem is referenced by:  eulerpartlemr  31627