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Theorem nexple 30612
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 479 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
2 simpl2 1248 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐵 ∈ ℝ)
3 simpl3 1250 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 2 ≤ 𝐵)
4 id 22 . . . . . . 7 (𝑘 = 1 → 𝑘 = 1)
5 oveq2 6918 . . . . . . 7 (𝑘 = 1 → (𝐵𝑘) = (𝐵↑1))
64, 5breq12d 4888 . . . . . 6 (𝑘 = 1 → (𝑘 ≤ (𝐵𝑘) ↔ 1 ≤ (𝐵↑1)))
76imbi2d 332 . . . . 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 6918 . . . . . . 7 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
108, 9breq12d 4888 . . . . . 6 (𝑘 = 𝑛 → (𝑘 ≤ (𝐵𝑘) ↔ 𝑛 ≤ (𝐵𝑛)))
1110imbi2d 332 . . . . 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 6918 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐵𝑘) = (𝐵↑(𝑛 + 1)))
1412, 13breq12d 4888 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝑘 ≤ (𝐵𝑘) ↔ (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))
1514imbi2d 332 . . . . 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 6918 . . . . . . 7 (𝑘 = 𝐴 → (𝐵𝑘) = (𝐵𝐴))
1816, 17breq12d 4888 . . . . . 6 (𝑘 = 𝐴 → (𝑘 ≤ (𝐵𝑘) ↔ 𝐴 ≤ (𝐵𝐴)))
1918imbi2d 332 . . . . 5 (𝑘 = 𝐴 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))))
20 simpl 476 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐵 ∈ ℝ)
21 1nn0 11643 . . . . . . 7 1 ∈ ℕ0
2221a1i 11 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℕ0)
23 1red 10364 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ∈ ℝ)
24 2re 11432 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ∈ ℝ)
26 1le2 11574 . . . . . . . 8 1 ≤ 2
2726a1i 11 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 2)
28 simpr 479 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 2 ≤ 𝐵)
2923, 25, 20, 27, 28letrd 10520 . . . . . 6 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ 𝐵)
3020, 22, 29expge1d 13328 . . . . 5 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1))
31 simp1 1170 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ)
3231nnnn0d 11685 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℕ0)
3332nn0red 11686 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℝ)
34 1red 10364 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ∈ ℝ)
3533, 34readdcld 10393 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ∈ ℝ)
36203ad2ant2 1168 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℝ)
3733, 36remulcld 10394 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ∈ ℝ)
3836, 32reexpcld 13326 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵𝑛) ∈ ℝ)
3938, 36remulcld 10394 . . . . . . . . 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 10394 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ∈ ℝ)
4231nnge1d 11406 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 1 ≤ 𝑛)
4334, 33, 33, 42leadd2dd 10974 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛))
4433recnd 10392 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ∈ ℂ)
4544times2d 11609 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛))
4643, 45breqtrrd 4903 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2))
4732nn0ge0d 11688 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝑛)
48 simp2r 1261 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 2 ≤ 𝐵)
4940, 36, 33, 47, 48lemul2ad 11301 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵))
5035, 41, 37, 46, 49letrd 10520 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵))
51 0red 10367 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ∈ ℝ)
52 0le2 11467 . . . . . . . . . . . . 13 0 ≤ 2
5352a1i 11 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 2)
5451, 25, 20, 53, 28letrd 10520 . . . . . . . . . . 11 ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 0 ≤ 𝐵)
55543ad2ant2 1168 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 0 ≤ 𝐵)
56 simp3 1172 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝑛 ≤ (𝐵𝑛))
5733, 38, 36, 55, 56lemul1ad 11300 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵𝑛) · 𝐵))
5835, 37, 39, 50, 57letrd 10520 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ ((𝐵𝑛) · 𝐵))
5936recnd 10392 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → 𝐵 ∈ ℂ)
6059, 32expp1d 13310 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵𝑛) · 𝐵))
6158, 60breqtrrd 4903 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝑛 ≤ (𝐵𝑛)) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))
62613exp 1152 . . . . . 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 11377 . . . 4 (𝐴 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴)))
65643impib 1148 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
661, 2, 3, 65syl3anc 1494 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ≤ (𝐵𝐴))
67 0le1 10882 . . . 4 0 ≤ 1
6867a1i 11 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 0 ≤ 1)
69 simpr 479 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 = 0)
7069oveq2d 6926 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = (𝐵↑0))
71 simpl2 1248 . . . . . 6 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ)
7271recnd 10392 . . . . 5 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
7372exp0d 13303 . . . 4 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1)
7470, 73eqtrd 2861 . . 3 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵𝐴) = 1)
7568, 69, 743brtr4d 4907 . 2 (((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵𝐴))
76 elnn0 11627 . . . 4 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7776biimpi 208 . . 3 (𝐴 ∈ ℕ0 → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
78773ad2ant1 1167 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝐴 ∈ ℕ ∨ 𝐴 = 0))
7966, 75, 78mpjaodan 986 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wo 878  w3a 1111   = wceq 1656  wcel 2164   class class class wbr 4875  (class class class)co 6910  cr 10258  0cc0 10259  1c1 10260   + caddc 10262   · cmul 10264  cle 10399  cn 11357  2c2 11413  0cn0 11625  cexp 13161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-n0 11626  df-z 11712  df-uz 11976  df-seq 13103  df-exp 13162
This theorem is referenced by:  oddpwdc  30957
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