Theorem eulerpartlemr 34872

Description: Lemma for eulerpart 34880. (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 3918	. . . 4 ⊢ (ℎ ∈ (𝑇 ∩ 𝑅) ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅))
2	1	anbi1i 636	. . 3 ⊢ ((ℎ ∈ (𝑇 ∩ 𝑅) ∧ ℎ ∈ 𝑃) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
3		elin 3918	. . 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 34863	. . . 4 ⊢ (ℎ ∈ 𝑂 ↔ (ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
8		cnveq 5857	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ◡𝑓 = ◡ℎ)
9	8	imaeq1d 6059	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (◡𝑓 “ ℕ) = (◡ℎ “ ℕ))
10	9	eleq1d 2847	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡ℎ “ ℕ) ∈ Fin))
11		fveq1 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (𝑓‘𝑘) = (ℎ‘𝑘))
12	11	oveq1d 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ((𝑓‘𝑘) · 𝑘) = ((ℎ‘𝑘) · 𝑘))
13	12	sumeq2sdv 15792	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘))
14	13	eqeq1d 2764	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
15	10, 14	anbi12d 644	. . . . . . . . . . . . . 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 3652	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ 𝑃 ↔ (ℎ ∈ (ℕ0 ↑m ℕ) ∧ ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁)))
17	16	simplbi 502	. . . . . . . . . . . 12 ⊢ (ℎ ∈ 𝑃 → ℎ ∈ (ℕ0 ↑m ℕ))
18		cnvimass 6082	. . . . . . . . . . . . 13 ⊢ (◡ℎ “ ℕ) ⊆ dom ℎ
19		nn0ex 12537	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ∈ V
20		nnex 12266	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
21	19, 20	elmap 8881	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) ↔ ℎ:ℕ⟶ℕ0)
22		fdm 6716	. . . . . . . . . . . . . 14 ⊢ (ℎ:ℕ⟶ℕ0 → dom ℎ = ℕ)
23	21, 22	sylbi 220	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) → dom ℎ = ℕ)
24	18, 23	sseqtrid 3976	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) → (◡ℎ “ ℕ) ⊆ ℕ)
25	17, 24	syl 18	. . . . . . . . . . 11 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ⊆ ℕ)
26	25	sselda 3934	. . . . . . . . . 10 ⊢ ((ℎ ∈ 𝑃 ∧ 𝑛 ∈ (◡ℎ “ ℕ)) → 𝑛 ∈ ℕ)
27	26	ralrimiva 3156	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ)
28	27	biantrurd 542	. . . . . . . 8 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)))
29	17	biantrurd 542	. . . . . . . 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 503	. . . . . . . . . 10 ⊢ (ℎ ∈ 𝑃 → ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
31	30	simpld 500	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ∈ Fin)
32	31	biantrud 541	. . . . . . . 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 308	. . . . . . 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 3923	. . . . . . . . . 10 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽)
35		breq2 5111	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
36	35	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
37		eulerpart.j	. . . . . . . . . . . 12 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
38	36, 37	elrab2 3652	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
39	38	ralbii 3110	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40		r19.26 3124	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
41	34, 39, 40	3bitri 300	. . . . . . . . 9 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
42	41	anbi2i 635	. . . . . . . 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 636	. . . . . . 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 292	. . . . . 6 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ ((ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin)))
45	9	sseq1d 3965	. . . . . . . 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 3652	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↔ (ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽))
48		vex 3457	. . . . . . . 8 ⊢ ℎ ∈ V
49		eulerpart.r	. . . . . . . 8 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
50	48, 10, 49	elab2 3639	. . . . . . 7 ⊢ (ℎ ∈ 𝑅 ↔ (◡ℎ “ ℕ) ∈ Fin)
51	47, 50	anbi12i 640	. . . . . 6 ⊢ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ↔ ((ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽) ∧ (◡ℎ “ ℕ) ∈ Fin))
52	44, 51	bitr4di 292	. . . . 5 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
53	52	pm5.32i 585	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛) ↔ (ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
54		ancom 466	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
55	7, 53, 54	3bitri 300	. . 3 ⊢ (ℎ ∈ 𝑂 ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
56	2, 3, 55	3bitr4ri 307	. 2 ⊢ (ℎ ∈ 𝑂 ↔ ℎ ∈ ((𝑇 ∩ 𝑅) ∩ 𝑃))
57	56	eqriv 2759	1 ⊢ 𝑂 = ((𝑇 ∩ 𝑅) ∩ 𝑃)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  {crab 3414   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560   class class class wbr 5107  {copab 5171   ↦ cmpt 5190  ^◡ccnv 5658  dom cdm 5659   ↾ cres 5661   “ cima 5662   ∘ ccom 5663  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418   supp csupp 8161   ↑_m cmap 8829  Fincfn 8955  1c1 11128   · cmul 11132   ≤ cle 11271  𝟭cind 12245  ℕcn 12260  2c2 12322  ℕ₀cn0 12531  ↑cexp 14127  Σcsu 15775   ∥ cdvds 16346  bitscbits 16513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-1cn 11185  ax-addcl 11187

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-map 8831  df-nn 12261  df-n0 12532  df-seq 14068  df-sum 15776

This theorem is used by: (None)