Theorem eulerpartlemo 34549

Description: Lemma for eulerpart 34566: 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 5815	. . . 4 ⊢ (𝑔 = 𝐴 → ◡𝑔 = ◡𝐴)
2	1	imaeq1d 6011	. . 3 ⊢ (𝑔 = 𝐴 → (◡𝑔 “ ℕ) = (◡𝐴 “ ℕ))
3	2	raleqdv 3297	. 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 3632	1 ⊢ (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391   class class class wbr 5072  ^◡ccnv 5617   “ cima 5621  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  Fincfn 8883  1c1 11030   · cmul 11034   ≤ cle 11171  ℕcn 12165  2c2 12227  ℕ₀cn0 12428  Σcsu 15639   ∥ cdvds 16212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631

This theorem is referenced by:  eulerpartlemr  34558