Theorem eulerpartlemt0 33368

Description: Lemma for eulerpart 33381. (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 5874	. . . . . 6 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴)
2	1	imaeq1d 6059	. . . . 5 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ))
3	2	sseq1d 4014	. . . 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 3687	. . 3 ⊢ (𝐴 ∈ 𝑇 ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
6	2	eleq1d 2819	. . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin))
7		eulerpart.r	. . . 4 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
8	6, 7	elab4g 3674	. . 3 ⊢ (𝐴 ∈ 𝑅 ↔ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin))
9	5, 8	anbi12i 628	. 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅) ↔ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin)))
10		elin 3965	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅))
11		elex 3493	. . . . 5 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴 ∈ V)
12	11	pm4.71i 561	. . . 4 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ 𝐴 ∈ V))
13	12	anbi1i 625	. . 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 1096	. . 3 ⊢ ((𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)))
15		an42 656	. . 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 303	. 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 303	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  {crab 3433  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  𝒫 cpw 4603   class class class wbr 5149  {copab 5211   ↦ cmpt 5232  ^◡ccnv 5676   “ cima 5680  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411   supp csupp 8146   ↑_m cmap 8820  Fincfn 8939  1c1 11111   · cmul 11115   ≤ cle 11249  ℕcn 12212  2c2 12267  ℕ₀cn0 12472  ↑cexp 14027  Σcsu 15632   ∥ cdvds 16197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690

This theorem is referenced by:  eulerpartlemf  33369  eulerpartlemt  33370  eulerpartlemmf  33374  eulerpartlemgvv  33375  eulerpartlemgu  33376  eulerpartlemgh  33377  eulerpartlemgs2  33379  eulerpartlemn  33380