Theorem nexple 32917

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 7375	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐵↑𝑘) = (𝐵↑1))
6	4, 5	breq12d 5098	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐵↑𝑘) = (𝐵↑𝑛))
10	8, 9	breq12d 5098	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → (𝐵↑𝑘) = (𝐵↑(𝑛 + 1)))
14	12, 13	breq12d 5098	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐵↑𝑘) = (𝐵↑𝐴))
18	16, 17	breq12d 5098	. . . . . 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 12453	. . . . . . 7 ⊢ 1 ∈ ℕ0
22	21	a1i 11	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23		1red 11145	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24		2re 12255	. . . . . . . 8 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26		1le2 12385	. . . . . . . 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 11303	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
30	20, 22, 29	expge1d 14127	. . . . 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 12189	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℝ)
33		1red 11145	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ∈ ℝ)
34	32, 33	readdcld 11174	. . . . . . . . 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 11175	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
37	31	nnnn0d 12498	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℕ0)
38	35, 37	reexpcld 14125	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑𝑛) ∈ ℝ)
39	38, 35	remulcld 11175	. . . . . . . . 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 11175	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ∈ ℝ)
42	31	nnge1d 12225	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ≤ 𝑛)
43	33, 32, 32, 42	leadd2dd 11765	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
44	32	recnd 11173	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℂ)
45	44	times2d 12421	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
46	43, 45	breqtrrd 5113	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
47	37	nn0ge0d 12501	. . . . . . . . . . 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 12096	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
50	34, 41, 36, 46, 49	letrd 11303	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51		0red 11147	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52		0le2 12283	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
53	52	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
54	51, 25, 20, 53, 28	letrd 11303	. . . . . . . . . . 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 12095	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵↑𝑛) · 𝐵))
58	34, 36, 39, 50, 57	letrd 11303	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ ((𝐵↑𝑛) · 𝐵))
59	35	recnd 11173	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝐵 ∈ ℂ)
60	59, 37	expp1d 14109	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵↑𝑛) · 𝐵))
61	58, 60	breqtrrd 5113	. . . . . . 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 12192	. . . 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 11673	. . . 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 7383	. . . 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 11173	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
73	72	exp0d 14102	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1)
74	70, 73	eqtrd 2771	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑𝐴) = 1)
75	68, 69, 74	3brtr4d 5117	. 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵↑𝐴))
76		elnn0 12439	. . . 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 5085  (class class class)co 7367  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   ≤ cle 11180  ℕcn 12174  2c2 12236  ℕ₀cn0 12437  ↑cexp 14023

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-z 12525  df-uz 12789  df-seq 13964  df-exp 14024

This theorem is referenced by:  oddpwdc  34498