Theorem eulerpartlemt0 31244

Description: Lemma for eulerpart 31257. (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 5630	. . . . . 6 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴)
2	1	imaeq1d 5805	. . . . 5 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ))
3	2	sseq1d 3919	. . . 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 3621	. . 3 ⊢ (𝐴 ∈ 𝑇 ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
6	2	eleq1d 2867	. . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin))
7		eulerpart.r	. . . 4 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
8	6, 7	elab4g 3609	. . 3 ⊢ (𝐴 ∈ 𝑅 ↔ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin))
9	5, 8	anbi12i 626	. 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅) ↔ ((𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (◡𝐴 “ ℕ) ∈ Fin)))
10		elin 4090	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑅))
11		elex 3455	. . . . 5 ⊢ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) → 𝐴 ∈ V)
12	11	pm4.71i 560	. . . 4 ⊢ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ 𝐴 ∈ V))
13	12	anbi1i 623	. . 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 1088	. . 3 ⊢ ((𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ ((◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)))
15		an42 653	. . 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 304	. 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 304	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  {cab 2775  ∀wral 3105  {crab 3109  Vcvv 3437   ∩ cin 3858   ⊆ wss 3859  ∅c0 4211  𝒫 cpw 4453   class class class wbr 4962  {copab 5024   ↦ cmpt 5041  ^◡ccnv 5442   “ cima 5446  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018   supp csupp 7681   ↑_𝑚 cmap 8256  Fincfn 8357  1c1 10384   · cmul 10388   ≤ cle 10522  ℕcn 11486  2c2 11540  ℕ₀cn0 11745  ↑cexp 13279  Σcsu 14876   ∥ cdvds 15440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456

This theorem is referenced by:  eulerpartlemf  31245  eulerpartlemt  31246  eulerpartlemmf  31250  eulerpartlemgvv  31251  eulerpartlemgu  31252  eulerpartlemgh  31253  eulerpartlemgs2  31255  eulerpartlemn  31256