Theorem eulerpartlemr 34615

Description: Lemma for eulerpart 34623. (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 3911	. . . 4 ⊢ (ℎ ∈ (𝑇 ∩ 𝑅) ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅))
2	1	anbi1i 632	. . 3 ⊢ ((ℎ ∈ (𝑇 ∩ 𝑅) ∧ ℎ ∈ 𝑃) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
3		elin 3911	. . 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 34606	. . . 4 ⊢ (ℎ ∈ 𝑂 ↔ (ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
8		cnveq 5834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ◡𝑓 = ◡ℎ)
9	8	imaeq1d 6034	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (◡𝑓 “ ℕ) = (◡ℎ “ ℕ))
10	9	eleq1d 2837	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡ℎ “ ℕ) ∈ Fin))
11		fveq1 6851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (𝑓‘𝑘) = (ℎ‘𝑘))
12	11	oveq1d 7396	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ((𝑓‘𝑘) · 𝑘) = ((ℎ‘𝑘) · 𝑘))
13	12	sumeq2sdv 15702	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘))
14	13	eqeq1d 2754	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
15	10, 14	anbi12d 640	. . . . . . . . . . . . . 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 3644	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ 𝑃 ↔ (ℎ ∈ (ℕ0 ↑m ℕ) ∧ ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁)))
17	16	simplbi 499	. . . . . . . . . . . 12 ⊢ (ℎ ∈ 𝑃 → ℎ ∈ (ℕ0 ↑m ℕ))
18		cnvimass 6057	. . . . . . . . . . . . 13 ⊢ (◡ℎ “ ℕ) ⊆ dom ℎ
19		nn0ex 12473	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ∈ V
20		nnex 12202	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
21	19, 20	elmap 8838	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) ↔ ℎ:ℕ⟶ℕ0)
22		fdm 6686	. . . . . . . . . . . . . 14 ⊢ (ℎ:ℕ⟶ℕ0 → dom ℎ = ℕ)
23	21, 22	sylbi 219	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) → dom ℎ = ℕ)
24	18, 23	sseqtrid 3969	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) → (◡ℎ “ ℕ) ⊆ ℕ)
25	17, 24	syl 17	. . . . . . . . . . 11 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ⊆ ℕ)
26	25	sselda 3927	. . . . . . . . . 10 ⊢ ((ℎ ∈ 𝑃 ∧ 𝑛 ∈ (◡ℎ “ ℕ)) → 𝑛 ∈ ℕ)
27	26	ralrimiva 3144	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ)
28	27	biantrurd 539	. . . . . . . 8 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)))
29	17	biantrurd 539	. . . . . . . 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 500	. . . . . . . . . 10 ⊢ (ℎ ∈ 𝑃 → ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
31	30	simpld 497	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ∈ Fin)
32	31	biantrud 538	. . . . . . . 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 307	. . . . . . 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 3916	. . . . . . . . . 10 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽)
35		breq2 5094	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
36	35	notbid 320	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
37		eulerpart.j	. . . . . . . . . . . 12 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
38	36, 37	elrab2 3644	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
39	38	ralbii 3098	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40		r19.26 3112	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
41	34, 39, 40	3bitri 299	. . . . . . . . 9 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
42	41	anbi2i 631	. . . . . . . 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 632	. . . . . . 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 291	. . . . . 6 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ ((ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin)))
45	9	sseq1d 3958	. . . . . . . 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 3644	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↔ (ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽))
48		vex 3448	. . . . . . . 8 ⊢ ℎ ∈ V
49		eulerpart.r	. . . . . . . 8 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
50	48, 10, 49	elab2 3632	. . . . . . 7 ⊢ (ℎ ∈ 𝑅 ↔ (◡ℎ “ ℕ) ∈ Fin)
51	47, 50	anbi12i 636	. . . . . 6 ⊢ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ↔ ((ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin))
52	44, 51	bitr4di 291	. . . . 5 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
53	52	pm5.32i 581	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛) ↔ (ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
54		ancom 463	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
55	7, 53, 54	3bitri 299	. . 3 ⊢ (ℎ ∈ 𝑂 ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
56	2, 3, 55	3bitr4ri 306	. 2 ⊢ (ℎ ∈ 𝑂 ↔ ℎ ∈ ((𝑇 ∩ 𝑅) ∩ 𝑃))
57	56	eqriv 2749	1 ⊢ 𝑂 = ((𝑇 ∩ 𝑅) ∩ 𝑃)

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1550   ∈ wcel 2132  {cab 2730  ∀wral 3066  {crab 3404   ∩ cin 3894   ⊆ wss 3895  ∅c0 4276  𝒫 cpw 4545   class class class wbr 5090  {copab 5152   ↦ cmpt 5171  ^◡ccnv 5635  dom cdm 5636   ↾ cres 5638   “ cima 5639   ∘ ccom 5640  ⟶wf 6502  ‘cfv 6506  (class class class)co 7381   ∈ cmpo 7383   supp csupp 8124   ↑_m cmap 8792  Fincfn 8912  1c1 11060   · cmul 11064   ≤ cle 11203  𝟭cind 12181  ℕcn 12196  2c2 12258  ℕ₀cn0 12467  ↑cexp 14060  Σcsu 15685   ∥ cdvds 16258  bitscbits 16425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-1cn 11117  ax-addcl 11119

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-map 8794  df-nn 12197  df-n0 12468  df-seq 14001  df-sum 15686

This theorem is referenced by: (None)