Theorem eulerpartlemt0 35071

Description: Lemma for eulerpart 35084. (Contributed by Thierry Arnoux, 19-Sep-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}
eulerpart.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h	⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m	⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
eulerpart.r	⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
eulerpart.t	⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}

Assertion

Ref	Expression
eulerpartlemt0	⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐽

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑧, 𝑔, 𝑘, 𝑛, 𝑟)   𝐷(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)   𝑃(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)   𝑅(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)   𝑇(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)   𝐹(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)   𝐻(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)   𝐽(𝑥, 𝑦, 𝑧, 𝑔, 𝑘, 𝑛, 𝑟)   𝑀(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)   𝑁(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)   𝑂(𝑥, 𝑦, 𝑧, 𝑓, 𝑔, 𝑘, 𝑛, 𝑟)

Proof of Theorem eulerpartlemt0

Step	Hyp	Ref	Expression
1		cnveq 5858	. . . . . 6 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴)
2	1	imaeq1d 6063	. . . . 5 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ))
3	2	sseq1d 3944	. . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ⊆ 𝐽 ↔ (◡𝐴 “ ℕ) ⊆ 𝐽))
4		eulerpart.t	. . . 4 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
5	3, 4	elrab2 3652	. . 3 ⊢ (𝐴 ∈ 𝑇 ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
6	2	eleq1d 2849	. . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin))
7		eulerpart.r	. . . 4 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
8	6, 7	elab4g 3640	. . 3 ⊢ (𝐴 ∈ 𝑅 ↔ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin))
9	5, 8	anbi12i 640	. 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin)))
10		elin 4096	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅))
11		elex 3475	. . . . 5 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴 ∈ V)
12	11	pm4.71i 569	. . . 4 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ 𝐴 ∈ V))
13	12	anbi1i 636	. . 3 ⊢ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ 𝐴 ∈ V) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)))
14		3anass 1111	. . 3 ⊢ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)))
15		an42 670	. . 3 ⊢ (((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin)) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ 𝐴 ∈ V) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)))
16	13, 14, 15	3bitr4i 306	. 2 ⊢ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin)))
17	9, 10, 16	3bitr4i 306	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560   class class class wbr 5106  {copab 5170   ↦ cmpt 5189  ^◡ccnv 5654   “ cima 5658  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445   supp csupp 8204   ↑_m cmap 8875  Fincfn 9001  1c1 11228   · cmul 11232   ≤ cle 11371  ℕcn 12362  2c2 12424  ℕ₀cn0 12633  ↑cexp 14231  Σcsu 15882   ∥ cdvds 16451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  eulerpartlemf  35072  eulerpartlemt  35073  eulerpartlemmf  35077  eulerpartlemgvv  35078  eulerpartlemgu  35079  eulerpartlemgh  35080  eulerpartlemgs2  35082  eulerpartlemn  35083