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Theorem nexple 30199
Description: A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
Assertion
Ref Expression
nexple ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))

Proof of Theorem nexple
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
2 simpl2 1085 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐵 ∈ ℝ)
3 simpl3 1086 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 2 ≤ 𝐵)
4 id 22 . . . . . . 7 (𝑘 = 1 → 𝑘 = 1)
5 oveq2 6698 . . . . . . 7 (𝑘 = 1 → (𝐵𝑘) = (𝐵↑1))
64, 5breq12d 4698 . . . . . 6 (𝑘 = 1 → (𝑘 ≤ (𝐵𝑘) ↔ 1 ≤ (𝐵↑1)))
76imbi2d 329 . . . . 5 (𝑘 = 1 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))))
8 id 22 . . . . . . 7 (𝑘 = 𝑛𝑘 = 𝑛)
9 oveq2 6698 . . . . . . 7 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
108, 9breq12d 4698 . . . . . 6 (𝑘 = 𝑛 → (𝑘 ≤ (𝐵𝑘) ↔ 𝑛 ≤ (𝐵𝑛)))
1110imbi2d 329 . . . . 5 (𝑘 = 𝑛 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑛 ≤ (𝐵𝑛))))
12 id 22 . . . . . . 7 (𝑘 = (𝑛 + 1) → 𝑘 = (𝑛 + 1))
13 oveq2 6698 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐵𝑘) = (𝐵↑(𝑛 + 1)))
1412, 13breq12d 4698 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝑘 ≤ (𝐵𝑘) ↔ (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))
1514imbi2d 329 . . . . 5 (𝑘 = (𝑛 + 1) → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))))
16 id 22 . . . . . . 7 (𝑘 = 𝐴𝑘 = 𝐴)
17 oveq2 6698 . . . . . . 7 (𝑘 = 𝐴 → (𝐵𝑘) = (𝐵𝐴))
1816, 17breq12d 4698 . . . . . 6 (𝑘 = 𝐴 → (𝑘 ≤ (𝐵𝑘) ↔ 𝐴 ≤ (𝐵𝐴)))
1918imbi2d 329 . . . . 5 (𝑘 = 𝐴 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))))
20 simpl 472 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐵 ∈ ℝ)
21 1nn0 11346 . . . . . . 7 1 ∈ ℕ0
2221a1i 11 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23 1red 10093 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24 2re 11128 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26 1le2 11279 . . . . . . . 8 1 ≤ 2
2726a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 2)
28 simpr 476 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ≤ 𝐵)
2923, 25, 20, 27, 28letrd 10232 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
3020, 22, 29expge1d 13067 . . . . 5 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))
31 simp1 1081 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ)
3231nnnn0d 11389 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ0)
3332nn0red 11390 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℝ)
34 1red 10093 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ∈ ℝ)
3533, 34readdcld 10107 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ∈ ℝ)
36203ad2ant2 1103 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℝ)
3733, 36remulcld 10108 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
3836, 32reexpcld 13065 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵𝑛) ∈ ℝ)
3938, 36remulcld 10108 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → ((𝐵𝑛) · 𝐵) ∈ ℝ)
4024a1i 11 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 2 ∈ ℝ)
4133, 40remulcld 10108 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ∈ ℝ)
4231nnge1d 11101 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ≤ 𝑛)
4334, 33, 33, 42leadd2dd 10680 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
4433recnd 10106 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℂ)
4544times2d 11314 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
4643, 45breqtrrd 4713 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
4732nn0ge0d 11392 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝑛)
48 simp2r 1108 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 2 ≤ 𝐵)
4940, 36, 33, 47, 48lemul2ad 11002 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
5035, 41, 37, 46, 49letrd 10232 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51 0red 10079 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52 0le2 11149 . . . . . . . . . . . . 13 0 ≤ 2
5352a1i 11 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
5451, 25, 20, 53, 28letrd 10232 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 𝐵)
55543ad2ant2 1103 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝐵)
56 simp3 1083 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ≤ (𝐵𝑛))
5733, 38, 36, 55, 56lemul1ad 11001 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵𝑛) · 𝐵))
5835, 37, 39, 50, 57letrd 10232 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ ((𝐵𝑛) · 𝐵))
5936recnd 10106 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℂ)
6059, 32expp1d 13049 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵𝑛) · 𝐵))
6158, 60breqtrrd 4713 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))
62613exp 1283 . . . . . 6 (𝑛 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝑛 ≤ (𝐵𝑛) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))))
6362a2d 29 . . . . 5 (𝑛 ∈ ℕ → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑛 ≤ (𝐵𝑛)) → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))))
647, 11, 15, 19, 30, 63nnind 11076 . . . 4 (𝐴 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴)))
65643impib 1281 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
661, 2, 3, 65syl3anc 1366 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ≤ (𝐵𝐴))
67 0le1 10589 . . . 4 0 ≤ 1
6867a1i 11 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 0 ≤ 1)
69 simpr 476 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 = 0)
7069oveq2d 6706 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = (𝐵↑0))
71 simpl2 1085 . . . . . 6 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ)
7271recnd 10106 . . . . 5 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
7372exp0d 13042 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1)
7470, 73eqtrd 2685 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = 1)
7568, 69, 743brtr4d 4717 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵𝐴))
76 elnn0 11332 . . . 4 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7776biimpi 206 . . 3 (𝐴 ∈ ℕ0 → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
78773ad2ant1 1102 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7966, 75, 78mpjaodan 844 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  cle 10113  cn 11058  2c2 11108  0cn0 11330  cexp 12900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-exp 12901
This theorem is referenced by:  oddpwdc  30544
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