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Theorem nexple 29195
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 475 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
2 simpl2 1057 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐵 ∈ ℝ)
3 simpl3 1058 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 2 ≤ 𝐵)
4 id 22 . . . . . . 7 (𝑘 = 1 → 𝑘 = 1)
5 oveq2 6434 . . . . . . 7 (𝑘 = 1 → (𝐵𝑘) = (𝐵↑1))
64, 5breq12d 4494 . . . . . 6 (𝑘 = 1 → (𝑘 ≤ (𝐵𝑘) ↔ 1 ≤ (𝐵↑1)))
76imbi2d 328 . . . . 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 6434 . . . . . . 7 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
108, 9breq12d 4494 . . . . . 6 (𝑘 = 𝑛 → (𝑘 ≤ (𝐵𝑘) ↔ 𝑛 ≤ (𝐵𝑛)))
1110imbi2d 328 . . . . 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 6434 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐵𝑘) = (𝐵↑(𝑛 + 1)))
1412, 13breq12d 4494 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝑘 ≤ (𝐵𝑘) ↔ (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))
1514imbi2d 328 . . . . 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 6434 . . . . . . 7 (𝑘 = 𝐴 → (𝐵𝑘) = (𝐵𝐴))
1816, 17breq12d 4494 . . . . . 6 (𝑘 = 𝐴 → (𝑘 ≤ (𝐵𝑘) ↔ 𝐴 ≤ (𝐵𝐴)))
1918imbi2d 328 . . . . 5 (𝑘 = 𝐴 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))))
20 simpl 471 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐵 ∈ ℝ)
21 1nn0 11063 . . . . . . 7 1 ∈ ℕ0
2221a1i 11 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23 1red 9810 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24 2re 10845 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26 1le2 10996 . . . . . . . 8 1 ≤ 2
2726a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 2)
28 simpr 475 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ≤ 𝐵)
2923, 25, 20, 27, 28letrd 9945 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
3020, 22, 29expge1d 12757 . . . . 5 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))
31 simp1 1053 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ)
3231nnnn0d 11106 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ0)
3332nn0red 11107 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℝ)
34 1red 9810 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ∈ ℝ)
3533, 34readdcld 9824 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ∈ ℝ)
36203ad2ant2 1075 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℝ)
3733, 36remulcld 9825 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
3836, 32reexpcld 12755 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵𝑛) ∈ ℝ)
3938, 36remulcld 9825 . . . . . . . . 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 9825 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ∈ ℝ)
4231nnge1d 10818 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ≤ 𝑛)
4334, 33, 33, 42leadd2dd 10391 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
4433recnd 9823 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℂ)
4544times2d 11031 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
4643, 45breqtrrd 4509 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
4732nn0ge0d 11109 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝑛)
48 simp2r 1080 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 2 ≤ 𝐵)
4940, 36, 33, 47, 48lemul2ad 10714 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
5035, 41, 37, 46, 49letrd 9945 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51 0red 9796 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52 0le2 10866 . . . . . . . . . . . . 13 0 ≤ 2
5352a1i 11 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
5451, 25, 20, 53, 28letrd 9945 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 𝐵)
55543ad2ant2 1075 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝐵)
56 simp3 1055 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ≤ (𝐵𝑛))
5733, 38, 36, 55, 56lemul1ad 10713 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵𝑛) · 𝐵))
5835, 37, 39, 50, 57letrd 9945 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ ((𝐵𝑛) · 𝐵))
5936recnd 9823 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℂ)
6059, 32expp1d 12739 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵𝑛) · 𝐵))
6158, 60breqtrrd 4509 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))
62613exp 1255 . . . . . 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 10793 . . . 4 (𝐴 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴)))
65643impib 1253 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
661, 2, 3, 65syl3anc 1317 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ≤ (𝐵𝐴))
67 0le1 10300 . . . 4 0 ≤ 1
6867a1i 11 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 0 ≤ 1)
69 simpr 475 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 = 0)
7069oveq2d 6442 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = (𝐵↑0))
71 simpl2 1057 . . . . . 6 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ)
7271recnd 9823 . . . . 5 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
7372exp0d 12732 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1)
7470, 73eqtrd 2548 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = 1)
7568, 69, 743brtr4d 4513 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵𝐴))
76 elnn0 11049 . . . 4 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7776biimpi 204 . . 3 (𝐴 ∈ ℕ0 → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
78773ad2ant1 1074 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7966, 75, 78mpjaodan 822 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1938   class class class wbr 4481  (class class class)co 6426  cr 9690  0cc0 9691  1c1 9692   + caddc 9694   · cmul 9696  cle 9830  cn 10775  2c2 10825  0cn0 11047  cexp 12590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-addrcl 9752  ax-mulcl 9753  ax-mulrcl 9754  ax-mulcom 9755  ax-addass 9756  ax-mulass 9757  ax-distr 9758  ax-i2m1 9759  ax-1ne0 9760  ax-1rid 9761  ax-rnegex 9762  ax-rrecex 9763  ax-cnre 9764  ax-pre-lttri 9765  ax-pre-lttrn 9766  ax-pre-ltadd 9767  ax-pre-mulgt0 9768
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-om 6834  df-2nd 6935  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-er 7505  df-en 7718  df-dom 7719  df-sdom 7720  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-sub 10019  df-neg 10020  df-nn 10776  df-2 10834  df-n0 11048  df-z 11119  df-uz 11428  df-seq 12532  df-exp 12591
This theorem is referenced by:  oddpwdc  29550
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