Theorem eulerpartlemgc 34327

Description: Lemma for eulerpart 34347. (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 12367	. . . . 5 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 2 ∈ ℝ)
3		bitsss 16472	. . . . 5 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
4		simprr 772	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ (bits‘(𝐴‘𝑡)))
5	3, 4	sselid 4006	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ ℕ0)
6	2, 5	reexpcld 14213	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℝ)
7		simprl 770	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℕ)
8	7	nnred 12308	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ)
9	6, 8	remulcld 11320	. 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 34322	. . . . . . 7 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
13	12	simpld 494	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
14	13	ffvelcdmda 7118	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
15	14	adantrr 716	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℕ0)
16	15	nn0red 12614	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ)
17	16, 8	remulcld 11320	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
18	10, 11	eulerpartlemsf 34324	. . . . 5 ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0
19	18	ffvelcdmi 7117	. . . 4 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) ∈ ℕ0)
20	19	adantr 480	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑆‘𝐴) ∈ ℕ0)
21	20	nn0red 12614	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑆‘𝐴) ∈ ℝ)
22	14	nn0red 12614	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℝ)
23	22	adantrr 716	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ)
24	7	nnrpd 13097	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ+)
25	24	rprege0d 13106	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26		bitsfi 16483	. . . . . 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 4008	. . . . . 6 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 𝑖 ∈ ℕ0)
31	28, 30	reexpcld 14213	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑖) ∈ ℝ)
32		0le2 12395	. . . . . . 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 14214	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 0 ≤ (2↑𝑖))
35	4	snssd 4834	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → {𝑛} ⊆ (bits‘(𝐴‘𝑡)))
36	27, 31, 34, 35	fsumless 15844	. . . 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 11318	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℂ)
38		oveq2 7456	. . . . . 6 ⊢ (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
39	38	sumsn 15794	. . . . 5 ⊢ ((𝑛 ∈ (bits‘(𝐴‘𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
40	4, 37, 39	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41		bitsinv1 16488	. . . . 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 5197	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ≤ (𝐴‘𝑡))
44		lemul1a 12148	. . 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 1373	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴‘𝑡) · 𝑡))
46		fzfid 14024	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → (1...(𝑆‘𝐴)) ∈ Fin)
47		elfznn 13613	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑆‘𝐴)) → 𝑘 ∈ ℕ)
48		ffvelcdm 7115	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ℕ0)
49	13, 47, 48	syl2an 595	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → (𝐴‘𝑘) ∈ ℕ0)
50	49	nn0red 12614	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → (𝐴‘𝑘) ∈ ℝ)
51	47	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 𝑘 ∈ ℕ)
52	51	nnred 12308	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 𝑘 ∈ ℝ)
53	50, 52	remulcld 11320	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ)
54	53	adantlr 714	. . . . . . 7 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ)
55	49	nn0ge0d 12616	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ (𝐴‘𝑘))
56		0red 11293	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ∈ ℝ)
57	51	nngt0d 12342	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 < 𝑘)
58	56, 52, 57	ltled 11438	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ 𝑘)
59	50, 52, 55, 58	mulge0d 11867	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘))
60	59	adantlr 714	. . . . . . 7 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘))
61		fveq2 6920	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
62		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
63	61, 62	oveq12d 7466	. . . . . . 7 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
64		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → 𝑡 ∈ (1...(𝑆‘𝐴)))
65	46, 54, 60, 63, 64	fsumge1 15845	. . . . . 6 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
66	65	adantlr 714	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
67		eldif 3986	. . . . . . 7 ⊢ (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
68		nndiffz1 32791	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆‘𝐴))) = (ℤ≥‘((𝑆‘𝐴) + 1)))
69	68	eleq2d 2830	. . . . . . . . . . . . 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 574	. . . . . . . . . . 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 34325	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)
73	71, 72	sylbi 217	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) = 0)
74	73	oveq1d 7463	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = (0 · 𝑡))
75		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))))
76	75	eldifad 3988	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ)
77	76	nncnd 12309	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℂ)
78	77	mul02d 11488	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (0 · 𝑡) = 0)
79	74, 78	eqtrd 2780	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = 0)
80		fzfid 14024	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (1...(𝑆‘𝐴)) ∈ Fin)
81	80, 53, 59	fsumge0 15843	. . . . . . . . 9 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 0 ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
82	81	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 0 ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
83	79, 82	eqbrtrd 5188	. . . . . . 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 594	. . . . . 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 467	. . . . 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 812	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
87	10, 11	eulerpartlemsv3 34326	. . . . 5 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
88	87	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
89	86, 88	breqtrrd 5194	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴))
90	89	adantrr 716	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴))
91	9, 17, 21, 45, 90	letrd 11447	1 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  {csn 4648   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699   “ cima 5703  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Fincfn 9003  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189   ≤ cle 11325  ℕcn 12293  2c2 12348  ℕ₀cn0 12553  ℤ_≥cuz 12903  ...cfz 13567  ↑cexp 14112  Σcsu 15734  bitscbits 16465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-fz 13568  df-fzo 13712  df-fl 13843  df-mod 13921  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-dvds 16303  df-bits 16468

This theorem is referenced by: (None)