Theorem eulerpartlemgc 34522

Description: Lemma for eulerpart 34542. (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 12246	. . . . 5 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 2 ∈ ℝ)
3		bitsss 16386	. . . . 5 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
4		simprr 773	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ (bits‘(𝐴‘𝑡)))
5	3, 4	sselid 3920	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ ℕ0)
6	2, 5	reexpcld 14116	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℝ)
7		simprl 771	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℕ)
8	7	nnred 12180	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ)
9	6, 8	remulcld 11166	. 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 34517	. . . . . . 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 7030	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
15	14	adantrr 718	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℕ0)
16	15	nn0red 12490	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ)
17	16, 8	remulcld 11166	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
18	10, 11	eulerpartlemsf 34519	. . . . 5 ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0
19	18	ffvelcdmi 7029	. . . 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 12490	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑆‘𝐴) ∈ ℝ)
22	14	nn0red 12490	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℝ)
23	22	adantrr 718	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ)
24	7	nnrpd 12975	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ+)
25	24	rprege0d 12984	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26		bitsfi 16397	. . . . . 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 3922	. . . . . 6 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 𝑖 ∈ ℕ0)
31	28, 30	reexpcld 14116	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑖) ∈ ℝ)
32		0le2 12274	. . . . . . 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 14117	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 0 ≤ (2↑𝑖))
35	4	snssd 4753	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → {𝑛} ⊆ (bits‘(𝐴‘𝑡)))
36	27, 31, 34, 35	fsumless 15750	. . . 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 11164	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℂ)
38		oveq2 7368	. . . . . 6 ⊢ (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
39	38	sumsn 15699	. . . . 5 ⊢ ((𝑛 ∈ (bits‘(𝐴‘𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
40	4, 37, 39	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41		bitsinv1 16402	. . . . 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 5117	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ≤ (𝐴‘𝑡))
44		lemul1a 12000	. . 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 1376	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴‘𝑡) · 𝑡))
46		fzfid 13926	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → (1...(𝑆‘𝐴)) ∈ Fin)
47		elfznn 13498	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑆‘𝐴)) → 𝑘 ∈ ℕ)
48		ffvelcdm 7027	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ℕ0)
49	13, 47, 48	syl2an 597	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → (𝐴‘𝑘) ∈ ℕ0)
50	49	nn0red 12490	. . . . . . . . 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 12180	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 𝑘 ∈ ℝ)
53	50, 52	remulcld 11166	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ)
54	53	adantlr 716	. . . . . . 7 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ)
55	49	nn0ge0d 12492	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ (𝐴‘𝑘))
56		0red 11138	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ∈ ℝ)
57	51	nngt0d 12217	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 < 𝑘)
58	56, 52, 57	ltled 11285	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ 𝑘)
59	50, 52, 55, 58	mulge0d 11718	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘))
60	59	adantlr 716	. . . . . . 7 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘))
61		fveq2 6834	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
62		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
63	61, 62	oveq12d 7378	. . . . . . 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 15751	. . . . . 6 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
66	65	adantlr 716	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
67		eldif 3900	. . . . . . 7 ⊢ (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
68		nndiffz1 32874	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆‘𝐴))) = (ℤ≥‘((𝑆‘𝐴) + 1)))
69	68	eleq2d 2823	. . . . . . . . . . . . 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 34520	. . . . . . . . . . 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 7375	. . . . . . . . 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 3902	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ)
77	76	nncnd 12181	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℂ)
78	77	mul02d 11335	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (0 · 𝑡) = 0)
79	74, 78	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = 0)
80		fzfid 13926	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (1...(𝑆‘𝐴)) ∈ Fin)
81	80, 53, 59	fsumge0 15749	. . . . . . . . 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 5108	. . . . . . 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 596	. . . . . 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 813	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
87	10, 11	eulerpartlemsv3 34521	. . . . 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 5114	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴))
90	89	adantrr 718	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴))
91	9, 17, 21, 45, 90	letrd 11294	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 1542   ∈ wcel 2114  {cab 2715   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  {csn 4568   class class class wbr 5086   ↦ cmpt 5167  ^◡ccnv 5623   “ cima 5627  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766  Fincfn 8886  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034   ≤ cle 11171  ℕcn 12165  2c2 12227  ℕ₀cn0 12428  ℤ_≥cuz 12779  ...cfz 13452  ↑cexp 14014  Σcsu 15639  bitscbits 16379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-ico 13295  df-fz 13453  df-fzo 13600  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-rlim 15442  df-sum 15640  df-dvds 16213  df-bits 16382

This theorem is referenced by: (None)