Theorem eulerpartlemr 32974

Description: Lemma for eulerpart 32982. (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 3926	. . . 4 ⊢ (ℎ ∈ (𝑇 ∩ 𝑅) ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅))
2	1	anbi1i 624	. . 3 ⊢ ((ℎ ∈ (𝑇 ∩ 𝑅) ∧ ℎ ∈ 𝑃) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
3		elin 3926	. . 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 32965	. . . 4 ⊢ (ℎ ∈ 𝑂 ↔ (ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
8		cnveq 5829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ◡𝑓 = ◡ℎ)
9	8	imaeq1d 6012	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (◡𝑓 “ ℕ) = (◡ℎ “ ℕ))
10	9	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡ℎ “ ℕ) ∈ Fin))
11		fveq1 6841	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (𝑓‘𝑘) = (ℎ‘𝑘))
12	11	oveq1d 7372	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ((𝑓‘𝑘) · 𝑘) = ((ℎ‘𝑘) · 𝑘))
13	12	sumeq2sdv 15589	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘))
14	13	eqeq1d 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
15	10, 14	anbi12d 631	. . . . . . . . . . . . . 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 3648	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ 𝑃 ↔ (ℎ ∈ (ℕ0 ↑m ℕ) ∧ ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁)))
17	16	simplbi 498	. . . . . . . . . . . 12 ⊢ (ℎ ∈ 𝑃 → ℎ ∈ (ℕ0 ↑m ℕ))
18		cnvimass 6033	. . . . . . . . . . . . 13 ⊢ (◡ℎ “ ℕ) ⊆ dom ℎ
19		nn0ex 12419	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ∈ V
20		nnex 12159	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
21	19, 20	elmap 8809	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) ↔ ℎ:ℕ⟶ℕ0)
22		fdm 6677	. . . . . . . . . . . . . 14 ⊢ (ℎ:ℕ⟶ℕ0 → dom ℎ = ℕ)
23	21, 22	sylbi 216	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) → dom ℎ = ℕ)
24	18, 23	sseqtrid 3996	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) → (◡ℎ “ ℕ) ⊆ ℕ)
25	17, 24	syl 17	. . . . . . . . . . 11 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ⊆ ℕ)
26	25	sselda 3944	. . . . . . . . . 10 ⊢ ((ℎ ∈ 𝑃 ∧ 𝑛 ∈ (◡ℎ “ ℕ)) → 𝑛 ∈ ℕ)
27	26	ralrimiva 3143	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ)
28	27	biantrurd 533	. . . . . . . 8 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)))
29	17	biantrurd 533	. . . . . . . 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 497	. . . . . . . . . 10 ⊢ (ℎ ∈ 𝑃 → ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
31	30	simpld 495	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ∈ Fin)
32	31	biantrud 532	. . . . . . . 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 304	. . . . . . 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 3932	. . . . . . . . . 10 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽)
35		breq2 5109	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
36	35	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
37		eulerpart.j	. . . . . . . . . . . 12 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
38	36, 37	elrab2 3648	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
39	38	ralbii 3096	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40		r19.26 3114	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
41	34, 39, 40	3bitri 296	. . . . . . . . 9 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
42	41	anbi2i 623	. . . . . . . 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 624	. . . . . . 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	bitr4di 288	. . . . . 6 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ ((ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin)))
45	9	sseq1d 3975	. . . . . . . 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 3648	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↔ (ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽))
48		vex 3449	. . . . . . . 8 ⊢ ℎ ∈ V
49		eulerpart.r	. . . . . . . 8 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
50	48, 10, 49	elab2 3634	. . . . . . 7 ⊢ (ℎ ∈ 𝑅 ↔ (◡ℎ “ ℕ) ∈ Fin)
51	47, 50	anbi12i 627	. . . . . 6 ⊢ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ↔ ((ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin))
52	44, 51	bitr4di 288	. . . . 5 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
53	52	pm5.32i 575	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛) ↔ (ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
54		ancom 461	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
55	7, 53, 54	3bitri 296	. . 3 ⊢ (ℎ ∈ 𝑂 ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
56	2, 3, 55	3bitr4ri 303	. 2 ⊢ (ℎ ∈ 𝑂 ↔ ℎ ∈ ((𝑇 ∩ 𝑅) ∩ 𝑃))
57	56	eqriv 2733	1 ⊢ 𝑂 = ((𝑇 ∩ 𝑅) ∩ 𝑃)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064  {crab 3407   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560   class class class wbr 5105  {copab 5167   ↦ cmpt 5188  ^◡ccnv 5632  dom cdm 5633   ↾ cres 5635   “ cima 5636   ∘ ccom 5637  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359   supp csupp 8092   ↑_m cmap 8765  Fincfn 8883  1c1 11052   · cmul 11056   ≤ cle 11190  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ↑cexp 13967  Σcsu 15570   ∥ cdvds 16136  bitscbits 16299  𝟭cind 32609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-sum 15571

This theorem is referenced by: (None)