Theorem nexple 32932

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 1194	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐵 ∈ ℝ)
3		simpl3 1195	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 2 ≤ 𝐵)
4		id 22	. . . . . . 7 ⊢ (𝑘 = 1 → 𝑘 = 1)
5		oveq2 7368	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐵↑𝑘) = (𝐵↑1))
6	4, 5	breq12d 5099	. . . . . 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 7368	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐵↑𝑘) = (𝐵↑𝑛))
10	8, 9	breq12d 5099	. . . . . 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 7368	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → (𝐵↑𝑘) = (𝐵↑(𝑛 + 1)))
14	12, 13	breq12d 5099	. . . . . 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 7368	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐵↑𝑘) = (𝐵↑𝐴))
18	16, 17	breq12d 5099	. . . . . 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 12444	. . . . . . 7 ⊢ 1 ∈ ℕ0
22	21	a1i 11	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23		1red 11136	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24		2re 12246	. . . . . . . 8 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26		1le2 12376	. . . . . . . 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 11294	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
30	20, 22, 29	expge1d 14118	. . . . 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 12180	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℝ)
33		1red 11136	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ∈ ℝ)
34	32, 33	readdcld 11165	. . . . . . . . 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 11166	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
37	31	nnnn0d 12489	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℕ0)
38	35, 37	reexpcld 14116	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑𝑛) ∈ ℝ)
39	38, 35	remulcld 11166	. . . . . . . . 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 11166	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ∈ ℝ)
42	31	nnge1d 12216	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ≤ 𝑛)
43	33, 32, 32, 42	leadd2dd 11756	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
44	32	recnd 11164	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℂ)
45	44	times2d 12412	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
46	43, 45	breqtrrd 5114	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
47	37	nn0ge0d 12492	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 0 ≤ 𝑛)
48		simp2r 1202	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 2 ≤ 𝐵)
49	40, 35, 32, 47, 48	lemul2ad 12087	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
50	34, 41, 36, 46, 49	letrd 11294	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51		0red 11138	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52		0le2 12274	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
53	52	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
54	51, 25, 20, 53, 28	letrd 11294	. . . . . . . . . . 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 12086	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵↑𝑛) · 𝐵))
58	34, 36, 39, 50, 57	letrd 11294	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ ((𝐵↑𝑛) · 𝐵))
59	35	recnd 11164	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝐵 ∈ ℂ)
60	59, 37	expp1d 14100	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵↑𝑛) · 𝐵))
61	58, 60	breqtrrd 5114	. . . . . . 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 12183	. . . 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 1374	. 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ≤ (𝐵↑𝐴))
67		0le1 11664	. . . 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 7376	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑𝐴) = (𝐵↑0))
71		simpl2 1194	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ)
72	71	recnd 11164	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
73	72	exp0d 14093	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1)
74	70, 73	eqtrd 2772	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑𝐴) = 1)
75	68, 69, 74	3brtr4d 5118	. 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵↑𝐴))
76		elnn0 12430	. . . 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 1542   ∈ wcel 2114   class class class wbr 5086  (class class class)co 7360  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034   ≤ cle 11171  ℕcn 12165  2c2 12227  ℕ₀cn0 12428  ↑cexp 14014

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-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  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

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-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-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-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-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-z 12516  df-uz 12780  df-seq 13955  df-exp 14015

This theorem is referenced by:  oddpwdc  34514