Theorem nexple 31643

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 488	. . 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 7199	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐵↑𝑘) = (𝐵↑1))
6	4, 5	breq12d 5052	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ≤ (𝐵↑𝑘) ↔ 1 ≤ (𝐵↑1)))
7	6	imbi2d 344	. . . . 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 7199	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐵↑𝑘) = (𝐵↑𝑛))
10	8, 9	breq12d 5052	. . . . . 6 ⊢ (𝑘 = 𝑛 → (𝑘 ≤ (𝐵↑𝑘) ↔ 𝑛 ≤ (𝐵↑𝑛)))
11	10	imbi2d 344	. . . . 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 7199	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → (𝐵↑𝑘) = (𝐵↑(𝑛 + 1)))
14	12, 13	breq12d 5052	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (𝑘 ≤ (𝐵↑𝑘) ↔ (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))
15	14	imbi2d 344	. . . . 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 7199	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐵↑𝑘) = (𝐵↑𝐴))
18	16, 17	breq12d 5052	. . . . . 6 ⊢ (𝑘 = 𝐴 → (𝑘 ≤ (𝐵↑𝑘) ↔ 𝐴 ≤ (𝐵↑𝐴)))
19	18	imbi2d 344	. . . . 5 ⊢ (𝑘 = 𝐴 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))))
20		simpl 486	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐵 ∈ ℝ)
21		1nn0 12071	. . . . . . 7 ⊢ 1 ∈ ℕ0
22	21	a1i 11	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23		1red 10799	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24		2re 11869	. . . . . . . 8 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26		1le2 12004	. . . . . . . 8 ⊢ 1 ≤ 2
27	26	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 2)
28		simpr 488	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ≤ 𝐵)
29	23, 25, 20, 27, 28	letrd 10954	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
30	20, 22, 29	expge1d 13700	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))
31		simp1 1138	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℕ)
32	31	nnnn0d 12115	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℕ0)
33	32	nn0red 12116	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℝ)
34		1red 10799	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ∈ ℝ)
35	33, 34	readdcld 10827	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ∈ ℝ)
36	20	3ad2ant2 1136	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝐵 ∈ ℝ)
37	33, 36	remulcld 10828	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
38	36, 32	reexpcld 13698	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑𝑛) ∈ ℝ)
39	38, 36	remulcld 10828	. . . . . . . . 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	33, 40	remulcld 10828	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ∈ ℝ)
42	31	nnge1d 11843	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ≤ 𝑛)
43	34, 33, 33, 42	leadd2dd 11412	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
44	33	recnd 10826	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℂ)
45	44	times2d 12039	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
46	43, 45	breqtrrd 5067	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
47	32	nn0ge0d 12118	. . . . . . . . . . 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, 36, 33, 47, 48	lemul2ad 11737	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
50	35, 41, 37, 46, 49	letrd 10954	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51		0red 10801	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52		0le2 11897	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
53	52	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
54	51, 25, 20, 53, 28	letrd 10954	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 𝐵)
55	54	3ad2ant2 1136	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 0 ≤ 𝐵)
56		simp3 1140	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ≤ (𝐵↑𝑛))
57	33, 38, 36, 55, 56	lemul1ad 11736	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵↑𝑛) · 𝐵))
58	35, 37, 39, 50, 57	letrd 10954	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ ((𝐵↑𝑛) · 𝐵))
59	36	recnd 10826	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝐵 ∈ ℂ)
60	59, 32	expp1d 13682	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵↑𝑛) · 𝐵))
61	58, 60	breqtrrd 5067	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))
62	61	3exp 1121	. . . . . 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 11813	. . . 4 ⊢ (𝐴 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴)))
65	64	3impib 1118	. . 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 11320	. . . 4 ⊢ 0 ≤ 1
68	67	a1i 11	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 0 ≤ 1)
69		simpr 488	. . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 = 0)
70	69	oveq2d 7207	. . . 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 10826	. . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
73	72	exp0d 13675	. . . 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 5071	. 2 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵↑𝐴))
76		elnn0 12057	. . . 4 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
77	76	biimpi 219	. . 3 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
78	77	3ad2ant1 1135	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
79	66, 75, 78	mpjaodan 959	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   class class class wbr 5039  (class class class)co 7191  ℝcr 10693  0cc0 10694  1c1 10695   + caddc 10697   · cmul 10699   ≤ cle 10833  ℕcn 11795  2c2 11850  ℕ₀cn0 12055  ↑cexp 13600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-2 11858  df-n0 12056  df-z 12142  df-uz 12404  df-seq 13540  df-exp 13601

This theorem is referenced by:  oddpwdc  31987