Theorem eulerpartlemgc 29751

Description: Lemma for eulerpart 29771. (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 10967	. . . . 5 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 2 ∈ ℝ)
3		bitsss 14986	. . . . 5 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
4		simprr 792	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ (bits‘(𝐴‘𝑡)))
5	3, 4	sseldi 3566	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ ℕ0)
6	2, 5	reexpcld 12887	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℝ)
7		simprl 790	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℕ)
8	7	nnred 10912	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ)
9	6, 8	remulcld 9949	. 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 29746	. . . . . . 7 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
13	12	simpld 474	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
14	13	ffvelrnda 6267	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
15	14	adantrr 749	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℕ0)
16	15	nn0red 11229	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ)
17	16, 8	remulcld 9949	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
18	10, 11	eulerpartlemsf 29748	. . . . 5 ⊢ 𝑆:((ℕ0 ↑𝑚 ℕ) ∩ 𝑅)⟶ℕ0
19	18	ffvelrni 6266	. . . 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 11229	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑆‘𝐴) ∈ ℝ)
22	14	nn0red 11229	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℝ)
23	22	adantrr 749	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ)
24	7	nnrpd 11746	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ+)
25	24	rprege0d 11755	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26		bitsfi 14997	. . . . . 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 3568	. . . . . 6 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 𝑖 ∈ ℕ0)
31	28, 30	reexpcld 12887	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑖) ∈ ℝ)
32		0le2 10988	. . . . . . 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 12888	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 0 ≤ (2↑𝑖))
35	4	snssd 4281	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → {𝑛} ⊆ (bits‘(𝐴‘𝑡)))
36	27, 31, 34, 35	fsumless 14369	. . . 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 9947	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℂ)
38		oveq2 6557	. . . . . 6 ⊢ (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
39	38	sumsn 14319	. . . . 5 ⊢ ((𝑛 ∈ (bits‘(𝐴‘𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
40	4, 37, 39	syl2anc 691	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41		bitsinv1 15002	. . . . 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 4614	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ≤ (𝐴‘𝑡))
44		lemul1a 10756	. . 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 1321	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴‘𝑡) · 𝑡))
46		fzfid 12634	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → (1...(𝑆‘𝐴)) ∈ Fin)
47		elfznn 12241	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑆‘𝐴)) → 𝑘 ∈ ℕ)
48		ffvelrn 6265	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ℕ0)
49	13, 47, 48	syl2an 493	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → (𝐴‘𝑘) ∈ ℕ0)
50	49	nn0red 11229	. . . . . . . . 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 10912	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 𝑘 ∈ ℝ)
53	50, 52	remulcld 9949	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ)
54	53	adantlr 747	. . . . . . 7 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ)
55	49	nn0ge0d 11231	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ (𝐴‘𝑘))
56		0red 9920	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ∈ ℝ)
57	51	nngt0d 10941	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 < 𝑘)
58	56, 52, 57	ltled 10064	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ 𝑘)
59	50, 52, 55, 58	mulge0d 10483	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘))
60	59	adantlr 747	. . . . . . 7 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘))
61		fveq2 6103	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
62		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
63	61, 62	oveq12d 6567	. . . . . . 7 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
64		simpr 476	. . . . . . 7 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → 𝑡 ∈ (1...(𝑆‘𝐴)))
65	46, 54, 60, 63, 64	fsumge1 14370	. . . . . 6 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
66	65	adantlr 747	. . . . 5 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
67		eldif 3550	. . . . . . 7 ⊢ (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
68		nndiffz1 28936	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆‘𝐴))) = (ℤ≥‘((𝑆‘𝐴) + 1)))
69	68	eleq2d 2673	. . . . . . . . . . . . 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 667	. . . . . . . . . . 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 29749	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)
73	71, 72	sylbi 206	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) = 0)
74	73	oveq1d 6564	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = (0 · 𝑡))
75		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))))
76	75	eldifad 3552	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ)
77	76	nncnd 10913	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℂ)
78	77	mul02d 10113	. . . . . . . . 9 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (0 · 𝑡) = 0)
79	74, 78	eqtrd 2644	. . . . . . . 8 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = 0)
80		fzfid 12634	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (1...(𝑆‘𝐴)) ∈ Fin)
81	80, 53, 59	fsumge0 14368	. . . . . . . . 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 4605	. . . . . . 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 492	. . . . . 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 678	. . . . 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 828	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
87	10, 11	eulerpartlemsv3 29750	. . . . 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 4611	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴))
90	89	adantrr 749	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴))
91	9, 17, 21, 45, 90	letrd 10073	1 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ℤ_≥cuz 11563  ...cfz 12197  ↑cexp 12722  Σcsu 14264  bitscbits 14979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-dvds 14822  df-bits 14982

This theorem is referenced by: (None)