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Theorem nexple 32833
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 484 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
2 simpl2 1193 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐵 ∈ ℝ)
3 simpl3 1194 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 2 ≤ 𝐵)
4 id 22 . . . . . . 7 (𝑘 = 1 → 𝑘 = 1)
5 oveq2 7439 . . . . . . 7 (𝑘 = 1 → (𝐵𝑘) = (𝐵↑1))
64, 5breq12d 5156 . . . . . 6 (𝑘 = 1 → (𝑘 ≤ (𝐵𝑘) ↔ 1 ≤ (𝐵↑1)))
76imbi2d 340 . . . . 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 7439 . . . . . . 7 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
108, 9breq12d 5156 . . . . . 6 (𝑘 = 𝑛 → (𝑘 ≤ (𝐵𝑘) ↔ 𝑛 ≤ (𝐵𝑛)))
1110imbi2d 340 . . . . 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 7439 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐵𝑘) = (𝐵↑(𝑛 + 1)))
1412, 13breq12d 5156 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝑘 ≤ (𝐵𝑘) ↔ (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))
1514imbi2d 340 . . . . 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 7439 . . . . . . 7 (𝑘 = 𝐴 → (𝐵𝑘) = (𝐵𝐴))
1816, 17breq12d 5156 . . . . . 6 (𝑘 = 𝐴 → (𝑘 ≤ (𝐵𝑘) ↔ 𝐴 ≤ (𝐵𝐴)))
1918imbi2d 340 . . . . 5 (𝑘 = 𝐴 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))))
20 simpl 482 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐵 ∈ ℝ)
21 1nn0 12542 . . . . . . 7 1 ∈ ℕ0
2221a1i 11 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23 1red 11262 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24 2re 12340 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26 1le2 12475 . . . . . . . 8 1 ≤ 2
2726a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 2)
28 simpr 484 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ≤ 𝐵)
2923, 25, 20, 27, 28letrd 11418 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
3020, 22, 29expge1d 14205 . . . . 5 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))
31 simp1 1137 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ)
3231nnred 12281 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℝ)
33 1red 11262 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ∈ ℝ)
3432, 33readdcld 11290 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ∈ ℝ)
35203ad2ant2 1135 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℝ)
3632, 35remulcld 11291 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
3731nnnn0d 12587 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ0)
3835, 37reexpcld 14203 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵𝑛) ∈ ℝ)
3938, 35remulcld 11291 . . . . . . . . 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)
4132, 40remulcld 11291 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ∈ ℝ)
4231nnge1d 12314 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ≤ 𝑛)
4333, 32, 32, 42leadd2dd 11878 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
4432recnd 11289 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℂ)
4544times2d 12510 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
4643, 45breqtrrd 5171 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
4737nn0ge0d 12590 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝑛)
48 simp2r 1201 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 2 ≤ 𝐵)
4940, 35, 32, 47, 48lemul2ad 12208 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
5034, 41, 36, 46, 49letrd 11418 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51 0red 11264 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52 0le2 12368 . . . . . . . . . . . . 13 0 ≤ 2
5352a1i 11 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
5451, 25, 20, 53, 28letrd 11418 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 𝐵)
55543ad2ant2 1135 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝐵)
56 simp3 1139 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ≤ (𝐵𝑛))
5732, 38, 35, 55, 56lemul1ad 12207 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵𝑛) · 𝐵))
5834, 36, 39, 50, 57letrd 11418 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ ((𝐵𝑛) · 𝐵))
5935recnd 11289 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℂ)
6059, 37expp1d 14187 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵𝑛) · 𝐵))
6158, 60breqtrrd 5171 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))
62613exp 1120 . . . . . 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 12284 . . . 4 (𝐴 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴)))
65643impib 1117 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
661, 2, 3, 65syl3anc 1373 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ≤ (𝐵𝐴))
67 0le1 11786 . . . 4 0 ≤ 1
6867a1i 11 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 0 ≤ 1)
69 simpr 484 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 = 0)
7069oveq2d 7447 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = (𝐵↑0))
71 simpl2 1193 . . . . . 6 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ)
7271recnd 11289 . . . . 5 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
7372exp0d 14180 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1)
7470, 73eqtrd 2777 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = 1)
7568, 69, 743brtr4d 5175 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵𝐴))
76 elnn0 12528 . . . 4 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7776biimpi 216 . . 3 (𝐴 ∈ ℕ0 → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
78773ad2ant1 1134 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7966, 75, 78mpjaodan 961 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  (class class class)co 7431  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  cle 11296  cn 12266  2c2 12321  0cn0 12526  cexp 14102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-exp 14103
This theorem is referenced by:  oddpwdc  34356
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