Theorem eulerpartlemr 32241

Description: Lemma for eulerpart 32249. (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 3899	. . . 4 ⊢ (ℎ ∈ (𝑇 ∩ 𝑅) ↔ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅))
2	1	anbi1i 623	. . 3 ⊢ ((ℎ ∈ (𝑇 ∩ 𝑅) ∧ ℎ ∈ 𝑃) ↔ ((ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅) ∧ ℎ ∈ 𝑃))
3		elin 3899	. . 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 32232	. . . 4 ⊢ (ℎ ∈ 𝑂 ↔ (ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛))
8		cnveq 5771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ◡𝑓 = ◡ℎ)
9	8	imaeq1d 5957	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (◡𝑓 “ ℕ) = (◡ℎ “ ℕ))
10	9	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡ℎ “ ℕ) ∈ Fin))
11		fveq1 6755	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (𝑓‘𝑘) = (ℎ‘𝑘))
12	11	oveq1d 7270	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → ((𝑓‘𝑘) · 𝑘) = ((ℎ‘𝑘) · 𝑘))
13	12	sumeq2sdv 15344	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘))
14	13	eqeq1d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
15	10, 14	anbi12d 630	. . . . . . . . . . . . . 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 3620	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ 𝑃 ↔ (ℎ ∈ (ℕ0 ↑m ℕ) ∧ ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁)))
17	16	simplbi 497	. . . . . . . . . . . 12 ⊢ (ℎ ∈ 𝑃 → ℎ ∈ (ℕ0 ↑m ℕ))
18		cnvimass 5978	. . . . . . . . . . . . 13 ⊢ (◡ℎ “ ℕ) ⊆ dom ℎ
19		nn0ex 12169	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ∈ V
20		nnex 11909	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
21	19, 20	elmap 8617	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (ℕ0 ↑m ℕ) ↔ ℎ:ℕ⟶ℕ0)
22		fdm 6593	. . . . . . . . . . . . . 14 ⊢ (ℎ:ℕ⟶ℕ0 → dom ℎ = ℕ)
23	21, 22	sylbi 216	. . . . . . . . . . . . 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 3917	. . . . . . . . . 10 ⊢ ((ℎ ∈ 𝑃 ∧ 𝑛 ∈ (◡ℎ “ ℕ)) → 𝑛 ∈ ℕ)
27	26	ralrimiva 3107	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ)
28	27	biantrurd 532	. . . . . . . 8 ⊢ (ℎ ∈ 𝑃 → (∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛)))
29	17	biantrurd 532	. . . . . . . 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 496	. . . . . . . . . 10 ⊢ (ℎ ∈ 𝑃 → ((◡ℎ “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((ℎ‘𝑘) · 𝑘) = 𝑁))
31	30	simpld 494	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑃 → (◡ℎ “ ℕ) ∈ Fin)
32	31	biantrud 531	. . . . . . . 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 3905	. . . . . . . . . 10 ⊢ ((◡ℎ “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽)
35		breq2 5074	. . . . . . . . . . . . 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 3620	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
39	38	ralbii 3090	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (◡ℎ “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡ℎ “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
40		r19.26 3094	. . . . . . . . . 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 622	. . . . . . . 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 623	. . . . . . 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 3948	. . . . . . . 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 3620	. . . . . . 7 ⊢ (ℎ ∈ 𝑇 ↔ (ℎ ∈ (ℕ0 ↑m ℕ) ∧ (◡ℎ “ ℕ) ⊆ 𝐽))
48		vex 3426	. . . . . . . 8 ⊢ ℎ ∈ V
49		eulerpart.r	. . . . . . . 8 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
50	48, 10, 49	elab2 3606	. . . . . . 7 ⊢ (ℎ ∈ 𝑅 ↔ (◡ℎ “ ℕ) ∈ Fin)
51	47, 50	anbi12i 626	. . . . . 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 574	. . . 4 ⊢ ((ℎ ∈ 𝑃 ∧ ∀𝑛 ∈ (◡ℎ “ ℕ) ¬ 2 ∥ 𝑛) ↔ (ℎ ∈ 𝑃 ∧ (ℎ ∈ 𝑇 ∧ ℎ ∈ 𝑅)))
54		ancom 460	. . . 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 2735	1 ⊢ 𝑂 = ((𝑇 ∩ 𝑅) ∩ 𝑃)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  {crab 3067   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530   class class class wbr 5070  {copab 5132   ↦ cmpt 5153  ^◡ccnv 5579  dom cdm 5580   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   supp csupp 7948   ↑_m cmap 8573  Fincfn 8691  1c1 10803   · cmul 10807   ≤ cle 10941  ℕcn 11903  2c2 11958  ℕ₀cn0 12163  ↑cexp 13710  Σcsu 15325   ∥ cdvds 15891  bitscbits 16054  𝟭cind 31878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-seq 13650  df-sum 15326

This theorem is referenced by: (None)