Theorem eulerpartlemo 34522

Description: Lemma for eulerpart 34539: 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 5822	. . . 4 ⊢ (𝑔 = 𝐴 → ◡𝑔 = ◡𝐴)
2	1	imaeq1d 6018	. . 3 ⊢ (𝑔 = 𝐴 → (◡𝑔 “ ℕ) = (◡𝐴 “ ℕ))
3	2	raleqdv 3296	. 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 3649	1 ⊢ (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399   class class class wbr 5098  ^◡ccnv 5623   “ cima 5627  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  Fincfn 8883  1c1 11027   · cmul 11031   ≤ cle 11167  ℕcn 12145  2c2 12200  ℕ₀cn0 12401  Σcsu 15609   ∥ cdvds 16179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  eulerpartlemr  34531