Theorem eulerpartlemgc 34620

Description: Lemma for eulerpart 34640. (Contributed by Thierry Arnoux, 9-Aug-2018.)

Hypotheses

Ref	Expression
eulerpartlems.r	⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
eulerpartlems.s	⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))

Assertion

Ref	Expression
eulerpartlemgc	⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆‘𝐴))

Distinct variable groups:   𝑓,𝑘,𝐴   𝑅,𝑓,𝑘   𝑡,𝑘,𝐴   𝑡,𝑅   𝑡,𝑆,𝑘

Allowed substitution hints:   𝐴(𝑛)   𝑅(𝑛)   𝑆(𝑓,𝑛)

Proof of Theorem eulerpartlemgc

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2re 12289	. . . . 5 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 2 ∈ ℝ)
3		bitsss 16443	. . . . 5 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
4		simprr 782	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ (bits‘(𝐴‘𝑡)))
5	3, 4	sselid 3934	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ ℕ0)
6	2, 5	reexpcld 14173	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℝ)
7		simprl 780	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℕ)
8	7	nnred 12222	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ)
9	6, 8	remulcld 11209	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℝ)
10		eulerpartlems.r	. . . . . . . 8 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
11		eulerpartlems.s	. . . . . . . 8 ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))
12	10, 11	eulerpartlemelr 34615	. . . . . . 7 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
13	12	simpld 498	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
14	13	ffvelcdmda 7061	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
15	14	adantrr 727	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℕ0)
16	15	nn0red 12540	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ)
17	16, 8	remulcld 11209	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
18	10, 11	eulerpartlemsf 34617	. . . . 5 ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0
19	18	ffvelcdmi 7060	. . . 4 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) ∈ ℕ0)
20	19	adantr 484	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑆‘𝐴) ∈ ℕ0)
21	20	nn0red 12540	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑆‘𝐴) ∈ ℝ)
22	14	nn0red 12540	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℝ)
23	22	adantrr 727	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ)
24	7	nnrpd 13032	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ+)
25	24	rprege0d 13041	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26		bitsfi 16454	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (bits‘(𝐴‘𝑡)) ∈ Fin)
27	15, 26	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (bits‘(𝐴‘𝑡)) ∈ Fin)
28	1	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 2 ∈ ℝ)
29	3	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (bits‘(𝐴‘𝑡)) ⊆ ℕ0)
30	29	sselda 3936	. . . . . 6 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 𝑖 ∈ ℕ0)
31	28, 30	reexpcld 14173	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑖) ∈ ℝ)
32		0le2 12317	. . . . . . 7 ⊢ 0 ≤ 2
33	32	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 0 ≤ 2)
34	28, 30, 33	expge0d 14174	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 0 ≤ (2↑𝑖))
35	4	snssd 4744	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → {𝑛} ⊆ (bits‘(𝐴‘𝑡)))
36	27, 31, 34, 35	fsumless 15807	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) ≤ Σ𝑖 ∈ (bits‘(𝐴‘𝑡))(2↑𝑖))
37	6	recnd 11207	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℂ)
38		oveq2 7400	. . . . . 6 ⊢ (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
39	38	sumsn 15756	. . . . 5 ⊢ ((𝑛 ∈ (bits‘(𝐴‘𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
40	4, 37, 39	syl2anc 593	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41		bitsinv1 16459	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → Σ𝑖 ∈ (bits‘(𝐴‘𝑡))(2↑𝑖) = (𝐴‘𝑡))
42	15, 41	syl 17	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ (bits‘(𝐴‘𝑡))(2↑𝑖) = (𝐴‘𝑡))
43	36, 40, 42	3brtr3d 5130	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ≤ (𝐴‘𝑡))
44		lemul1a 12042	. . 3 ⊢ ((((2↑𝑛) ∈ ℝ ∧ (𝐴‘𝑡) ∈ ℝ ∧ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡)) ∧ (2↑𝑛) ≤ (𝐴‘𝑡)) → ((2↑𝑛) · 𝑡) ≤ ((𝐴‘𝑡) · 𝑡))
45	6, 23, 25, 43, 44	syl31anc 1391	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴‘𝑡) · 𝑡))
46		fzfid 13983	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → (1...(𝑆‘𝐴)) ∈ Fin)
47		elfznn 13555	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑆‘𝐴)) → 𝑘 ∈ ℕ)
48		ffvelcdm 7058	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ℕ0)
49	13, 47, 48	syl2an 605	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → (𝐴‘𝑘) ∈ ℕ0)
50	49	nn0red 12540	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → (𝐴‘𝑘) ∈ ℝ)
51	47	adantl 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 𝑘 ∈ ℕ)
52	51	nnred 12222	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 𝑘 ∈ ℝ)
53	50, 52	remulcld 11209	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ)
54	53	adantlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ)
55	49	nn0ge0d 12542	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ (𝐴‘𝑘))
56		0red 11181	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ∈ ℝ)
57	51	nngt0d 12259	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 < 𝑘)
58	56, 52, 57	ltled 11328	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ 𝑘)
59	50, 52, 55, 58	mulge0d 11761	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘))
60	59	adantlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘))
61		fveq2 6863	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
62		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
63	61, 62	oveq12d 7410	. . . . . . 7 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
64		simpr 488	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → 𝑡 ∈ (1...(𝑆‘𝐴)))
65	46, 54, 60, 63, 64	fsumge1 15808	. . . . . 6 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
66	65	adantlr 725	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
67		eldif 3914	. . . . . . 7 ⊢ (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
68		nndiffz1 32938	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆‘𝐴))) = (ℤ≥‘((𝑆‘𝐴) + 1)))
69	68	eleq2d 2847	. . . . . . . . . . . . 13 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
70	19, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
71	70	pm5.32i 582	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) ↔ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
72	10, 11	eulerpartlems 34618	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)
73	71, 72	sylbi 219	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) = 0)
74	73	oveq1d 7407	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = (0 · 𝑡))
75		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))))
76	75	eldifad 3916	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ)
77	76	nncnd 12223	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℂ)
78	77	mul02d 11378	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (0 · 𝑡) = 0)
79	74, 78	eqtrd 2796	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = 0)
80		fzfid 13983	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (1...(𝑆‘𝐴)) ∈ Fin)
81	80, 53, 59	fsumge0 15806	. . . . . . . . 9 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 0 ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
82	81	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 0 ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
83	79, 82	eqbrtrd 5121	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
84	67, 83	sylan2br 604	. . . . . 6 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
85	84	anassrs 471	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
86	66, 85	pm2.61dan 822	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
87	10, 11	eulerpartlemsv3 34619	. . . . 5 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
88	87	adantr 484	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
89	86, 88	breqtrrd 5127	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴))
90	89	adantrr 727	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴))
91	9, 17, 21, 45, 90	letrd 11337	1 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {cab 2739   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  {csn 4581   class class class wbr 5099   ↦ cmpt 5180  ^◡ccnv 5644   “ cima 5648  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392   ↑_m cmap 8803  Fincfn 8923  ℂcc 11068  ℝcr 11069  0cc0 11070  1c1 11071   + caddc 11073   · cmul 11075   ≤ cle 11214  ℕcn 12207  2c2 12269  ℕ₀cn0 12478  ℤ_≥cuz 12836  ...cfz 13509  ↑cexp 14071  Σcsu 15696  bitscbits 16436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-supp 8136  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-pm 8806  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-sup 9385  df-inf 9386  df-oi 9455  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-rp 12991  df-ico 13352  df-fz 13510  df-fzo 13657  df-fl 13799  df-mod 13877  df-seq 14012  df-exp 14072  df-hash 14341  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-clim 15498  df-rlim 15499  df-sum 15697  df-dvds 16270  df-bits 16439

This theorem is referenced by: (None)