Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nexple Structured version   Visualization version   GIF version

Theorem nexple 31376
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 488 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
2 simpl2 1189 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐵 ∈ ℝ)
3 simpl3 1190 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 2 ≤ 𝐵)
4 id 22 . . . . . . 7 (𝑘 = 1 → 𝑘 = 1)
5 oveq2 7147 . . . . . . 7 (𝑘 = 1 → (𝐵𝑘) = (𝐵↑1))
64, 5breq12d 5046 . . . . . 6 (𝑘 = 1 → (𝑘 ≤ (𝐵𝑘) ↔ 1 ≤ (𝐵↑1)))
76imbi2d 344 . . . . 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 7147 . . . . . . 7 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
108, 9breq12d 5046 . . . . . 6 (𝑘 = 𝑛 → (𝑘 ≤ (𝐵𝑘) ↔ 𝑛 ≤ (𝐵𝑛)))
1110imbi2d 344 . . . . 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 7147 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐵𝑘) = (𝐵↑(𝑛 + 1)))
1412, 13breq12d 5046 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝑘 ≤ (𝐵𝑘) ↔ (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))
1514imbi2d 344 . . . . 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 7147 . . . . . . 7 (𝑘 = 𝐴 → (𝐵𝑘) = (𝐵𝐴))
1816, 17breq12d 5046 . . . . . 6 (𝑘 = 𝐴 → (𝑘 ≤ (𝐵𝑘) ↔ 𝐴 ≤ (𝐵𝐴)))
1918imbi2d 344 . . . . 5 (𝑘 = 𝐴 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))))
20 simpl 486 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐵 ∈ ℝ)
21 1nn0 11905 . . . . . . 7 1 ∈ ℕ0
2221a1i 11 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23 1red 10635 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24 2re 11703 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26 1le2 11838 . . . . . . . 8 1 ≤ 2
2726a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 2)
28 simpr 488 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ≤ 𝐵)
2923, 25, 20, 27, 28letrd 10790 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
3020, 22, 29expge1d 13529 . . . . 5 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))
31 simp1 1133 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ)
3231nnnn0d 11947 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ0)
3332nn0red 11948 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℝ)
34 1red 10635 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ∈ ℝ)
3533, 34readdcld 10663 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ∈ ℝ)
36203ad2ant2 1131 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℝ)
3733, 36remulcld 10664 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
3836, 32reexpcld 13527 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵𝑛) ∈ ℝ)
3938, 36remulcld 10664 . . . . . . . . 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 10664 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ∈ ℝ)
4231nnge1d 11677 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ≤ 𝑛)
4334, 33, 33, 42leadd2dd 11248 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
4433recnd 10662 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℂ)
4544times2d 11873 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
4643, 45breqtrrd 5061 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
4732nn0ge0d 11950 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝑛)
48 simp2r 1197 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 2 ≤ 𝐵)
4940, 36, 33, 47, 48lemul2ad 11573 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
5035, 41, 37, 46, 49letrd 10790 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51 0red 10637 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52 0le2 11731 . . . . . . . . . . . . 13 0 ≤ 2
5352a1i 11 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
5451, 25, 20, 53, 28letrd 10790 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 𝐵)
55543ad2ant2 1131 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝐵)
56 simp3 1135 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ≤ (𝐵𝑛))
5733, 38, 36, 55, 56lemul1ad 11572 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵𝑛) · 𝐵))
5835, 37, 39, 50, 57letrd 10790 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ ((𝐵𝑛) · 𝐵))
5936recnd 10662 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℂ)
6059, 32expp1d 13511 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵𝑛) · 𝐵))
6158, 60breqtrrd 5061 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))
62613exp 1116 . . . . . 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 11647 . . . 4 (𝐴 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴)))
65643impib 1113 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
661, 2, 3, 65syl3anc 1368 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ≤ (𝐵𝐴))
67 0le1 11156 . . . 4 0 ≤ 1
6867a1i 11 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 0 ≤ 1)
69 simpr 488 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 = 0)
7069oveq2d 7155 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = (𝐵↑0))
71 simpl2 1189 . . . . . 6 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ)
7271recnd 10662 . . . . 5 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
7372exp0d 13504 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1)
7470, 73eqtrd 2836 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = 1)
7568, 69, 743brtr4d 5065 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵𝐴))
76 elnn0 11891 . . . 4 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7776biimpi 219 . . 3 (𝐴 ∈ ℕ0 → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
78773ad2ant1 1130 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7966, 75, 78mpjaodan 956 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112   class class class wbr 5033  (class class class)co 7139  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535  cle 10669  cn 11629  2c2 11684  0cn0 11889  cexp 13429
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-seq 13369  df-exp 13430
This theorem is referenced by:  oddpwdc  31720
  Copyright terms: Public domain W3C validator