Theorem eulerpartlemr 29763

Description: Lemma for eulerpart 29771. (Contributed by Thierry Arnoux, 13-Nov-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}
eulerpart.g	⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))

Assertion

Ref	Expression
eulerpartlemr	⊢ 𝑂 = ((𝑇 ∩ 𝑅) ∩ 𝑃)

Distinct variable groups:   𝑓,𝑘,𝑛,𝑧   𝑓,𝐽,𝑛   𝑓,𝑁   𝑔,𝑛,𝑃

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

Proof of Theorem eulerpartlemr

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3758	. . . 4 ⊢ (ℎ ∈ (𝑇 ∩ 𝑅) ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅))
2	1	anbi1i 727	. . 3 ⊢ ((ℎ ∈ (𝑇 ∩ 𝑅) ∧ ℎ ∈ 𝑃) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
3		elin 3758	. . 3 ⊢ (ℎ ∈ ((𝑇 ∩ 𝑅) ∩ 𝑃) ↔ (ℎ ∈ (𝑇 ∩ 𝑅) ∧ ℎ ∈ 𝑃))
4		eulerpart.p	. . . . 5 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
5		eulerpart.o	. . . . 5 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
6		eulerpart.d	. . . . 5 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
7	4, 5, 6	eulerpartlemo 29754	. . . 4 ⊢ (ℎ ∈ 𝑂 ↔ (ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
8		cnveq 5218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ◡𝑓 = ◡ℎ)
9	8	imaeq1d 5384	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (◡𝑓 “ ℕ) = (◡ℎ “ ℕ))
10	9	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡ℎ “ ℕ) ∈ Fin))
11		fveq1 6102	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (𝑓‘𝑘) = (ℎ‘𝑘))
12	11	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ((𝑓‘𝑘) · 𝑘) = ((ℎ‘𝑘) · 𝑘))
13	12	sumeq2sdv 14282	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘))
14	13	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
15	10, 14	anbi12d 743	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁) ↔ ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁)))
16	15, 4	elrab2 3333	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ 𝑃 ↔ (ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁)))
17	16	simplbi 475	. . . . . . . . . . . 12 ⊢ (ℎ ∈ 𝑃 → ℎ ∈ (ℕ0 ↑𝑚 ℕ))
18		cnvimass 5404	. . . . . . . . . . . . 13 ⊢ (◡ℎ “ ℕ) ⊆ dom ℎ
19		nn0ex 11175	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ∈ V
20		nnex 10903	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
21	19, 20	elmap 7772	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (ℕ0 ↑𝑚 ℕ) ↔ ℎ:ℕ⟶ℕ0)
22		fdm 5964	. . . . . . . . . . . . . 14 ⊢ (ℎ:ℕ⟶ℕ0 → dom ℎ = ℕ)
23	21, 22	sylbi 206	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ (ℕ0 ↑𝑚 ℕ) → dom ℎ = ℕ)
24	18, 23	syl5sseq 3616	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (ℕ0 ↑𝑚 ℕ) → (◡ℎ “ ℕ) ⊆ ℕ)
25	17, 24	syl 17	. . . . . . . . . . 11 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ⊆ ℕ)
26	25	sselda 3568	. . . . . . . . . 10 ⊢ ((ℎ ∈ 𝑃 ∧ 𝑛 ∈ (◡ℎ “ ℕ)) → 𝑛 ∈ ℕ)
27	26	ralrimiva 2949	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ)
28	27	biantrurd 528	. . . . . . . 8 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)))
29	17	biantrurd 528	. . . . . . . 8 ⊢ (ℎ ∈ 𝑃 → ((∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛) ↔ (ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))))
30	16	simprbi 479	. . . . . . . . . 10 ⊢ (ℎ ∈ 𝑃 → ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
31	30	simpld 474	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ∈ Fin)
32	31	biantrud 527	. . . . . . . 8 ⊢ (ℎ ∈ 𝑃 → ((ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)) ↔ ((ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)) ∧ (◡ℎ “ ℕ) ∈ Fin)))
33	28, 29, 32	3bitrd 293	. . . . . . 7 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ ((ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)) ∧ (◡ℎ “ ℕ) ∈ Fin)))
34		dfss3 3558	. . . . . . . . . 10 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽)
35		breq2 4587	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
36	35	notbid 307	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
37		eulerpart.j	. . . . . . . . . . . 12 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
38	36, 37	elrab2 3333	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
39	38	ralbii 2963	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40		r19.26 3046	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
41	34, 39, 40	3bitri 285	. . . . . . . . 9 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
42	41	anbi2i 726	. . . . . . . 8 ⊢ ((ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ↔ (ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)))
43	42	anbi1i 727	. . . . . . 7 ⊢ (((ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin) ↔ ((ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)) ∧ (◡ℎ “ ℕ) ∈ Fin))
44	33, 43	syl6bbr 277	. . . . . 6 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ ((ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin)))
45	9	sseq1d 3595	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((◡𝑓 “ ℕ) ⊆ 𝐽 ↔ (◡ℎ “ ℕ) ⊆ 𝐽))
46		eulerpart.t	. . . . . . . 8 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
47	45, 46	elrab2 3333	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↔ (ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽))
48		vex 3176	. . . . . . . 8 ⊢ ℎ ∈ V
49		eulerpart.r	. . . . . . . 8 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
50	48, 10, 49	elab2 3323	. . . . . . 7 ⊢ (ℎ ∈ 𝑅 ↔ (◡ℎ “ ℕ) ∈ Fin)
51	47, 50	anbi12i 729	. . . . . 6 ⊢ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ↔ ((ℎ ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin))
52	44, 51	syl6bbr 277	. . . . 5 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
53	52	pm5.32i 667	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛) ↔ (ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
54		ancom 465	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
55	7, 53, 54	3bitri 285	. . 3 ⊢ (ℎ ∈ 𝑂 ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
56	2, 3, 55	3bitr4ri 292	. 2 ⊢ (ℎ ∈ 𝑂 ↔ ℎ ∈ ((𝑇 ∩ 𝑅) ∩ 𝑃))
57	56	eqriv 2607	1 ⊢ 𝑂 = ((𝑇 ∩ 𝑅) ∩ 𝑃)

Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   supp csupp 7182   ↑_𝑚 cmap 7744  Fincfn 7841  1c1 9816   · cmul 9820   ≤ cle 9954  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ↑cexp 12722  Σcsu 14264   ∥ cdvds 14821  bitscbits 14979  𝟭cind 29400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-sum 14265

This theorem is referenced by: (None)