Theorem nexple 32833

Description: A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)

Assertion

Ref	Expression
nexple	⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))

Proof of Theorem nexple

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
2		simpl2 1193	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐵 ∈ ℝ)
3		simpl3 1194	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 2 ≤ 𝐵)
4		id 22	. . . . . . 7 ⊢ (𝑘 = 1 → 𝑘 = 1)
5		oveq2 7439	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐵↑𝑘) = (𝐵↑1))
6	4, 5	breq12d 5156	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ≤ (𝐵↑𝑘) ↔ 1 ≤ (𝐵↑1)))
7	6	imbi2d 340	. . . . 5 ⊢ (𝑘 = 1 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))))
8		id 22	. . . . . . 7 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
9		oveq2 7439	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐵↑𝑘) = (𝐵↑𝑛))
10	8, 9	breq12d 5156	. . . . . 6 ⊢ (𝑘 = 𝑛 → (𝑘 ≤ (𝐵↑𝑘) ↔ 𝑛 ≤ (𝐵↑𝑛)))
11	10	imbi2d 340	. . . . 5 ⊢ (𝑘 = 𝑛 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑛 ≤ (𝐵↑𝑛))))
12		id 22	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → 𝑘 = (𝑛 + 1))
13		oveq2 7439	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → (𝐵↑𝑘) = (𝐵↑(𝑛 + 1)))
14	12, 13	breq12d 5156	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (𝑘 ≤ (𝐵↑𝑘) ↔ (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))
15	14	imbi2d 340	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))))
16		id 22	. . . . . . 7 ⊢ (𝑘 = 𝐴 → 𝑘 = 𝐴)
17		oveq2 7439	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐵↑𝑘) = (𝐵↑𝐴))
18	16, 17	breq12d 5156	. . . . . 6 ⊢ (𝑘 = 𝐴 → (𝑘 ≤ (𝐵↑𝑘) ↔ 𝐴 ≤ (𝐵↑𝐴)))
19	18	imbi2d 340	. . . . 5 ⊢ (𝑘 = 𝐴 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))))
20		simpl 482	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐵 ∈ ℝ)
21		1nn0 12542	. . . . . . 7 ⊢ 1 ∈ ℕ0
22	21	a1i 11	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23		1red 11262	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24		2re 12340	. . . . . . . 8 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26		1le2 12475	. . . . . . . 8 ⊢ 1 ≤ 2
27	26	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 2)
28		simpr 484	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ≤ 𝐵)
29	23, 25, 20, 27, 28	letrd 11418	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
30	20, 22, 29	expge1d 14205	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))
31		simp1 1137	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℕ)
32	31	nnred 12281	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℝ)
33		1red 11262	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ∈ ℝ)
34	32, 33	readdcld 11290	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ∈ ℝ)
35	20	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝐵 ∈ ℝ)
36	32, 35	remulcld 11291	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
37	31	nnnn0d 12587	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℕ0)
38	35, 37	reexpcld 14203	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑𝑛) ∈ ℝ)
39	38, 35	remulcld 11291	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → ((𝐵↑𝑛) · 𝐵) ∈ ℝ)
40	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 2 ∈ ℝ)
41	32, 40	remulcld 11291	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ∈ ℝ)
42	31	nnge1d 12314	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ≤ 𝑛)
43	33, 32, 32, 42	leadd2dd 11878	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
44	32	recnd 11289	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℂ)
45	44	times2d 12510	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
46	43, 45	breqtrrd 5171	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
47	37	nn0ge0d 12590	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 0 ≤ 𝑛)
48		simp2r 1201	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 2 ≤ 𝐵)
49	40, 35, 32, 47, 48	lemul2ad 12208	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
50	34, 41, 36, 46, 49	letrd 11418	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51		0red 11264	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52		0le2 12368	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
53	52	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
54	51, 25, 20, 53, 28	letrd 11418	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 𝐵)
55	54	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 0 ≤ 𝐵)
56		simp3 1139	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ≤ (𝐵↑𝑛))
57	32, 38, 35, 55, 56	lemul1ad 12207	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵↑𝑛) · 𝐵))
58	34, 36, 39, 50, 57	letrd 11418	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ ((𝐵↑𝑛) · 𝐵))
59	35	recnd 11289	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝐵 ∈ ℂ)
60	59, 37	expp1d 14187	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵↑𝑛) · 𝐵))
61	58, 60	breqtrrd 5171	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))
62	61	3exp 1120	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝑛 ≤ (𝐵↑𝑛) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))))
63	62	a2d 29	. . . . 5 ⊢ (𝑛 ∈ ℕ → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑛 ≤ (𝐵↑𝑛)) → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))))
64	7, 11, 15, 19, 30, 63	nnind 12284	. . . 4 ⊢ (𝐴 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴)))
65	64	3impib 1117	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))
66	1, 2, 3, 65	syl3anc 1373	. 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ≤ (𝐵↑𝐴))
67		0le1 11786	. . . 4 ⊢ 0 ≤ 1
68	67	a1i 11	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 0 ≤ 1)
69		simpr 484	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 = 0)
70	69	oveq2d 7447	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑𝐴) = (𝐵↑0))
71		simpl2 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ)
72	71	recnd 11289	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
73	72	exp0d 14180	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1)
74	70, 73	eqtrd 2777	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑𝐴) = 1)
75	68, 69, 74	3brtr4d 5175	. 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵↑𝐴))
76		elnn0 12528	. . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
77	76	biimpi 216	. . 3 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
78	77	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
79	66, 75, 78	mpjaodan 961	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   class class class wbr 5143  (class class class)co 7431  ℝcr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   ≤ cle 11296  ℕcn 12266  2c2 12321  ℕ₀cn0 12526  ↑cexp 14102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-exp 14103

This theorem is referenced by:  oddpwdc  34356