MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  leexp2r Structured version   Visualization version   GIF version

Theorem leexp2r 14214
Description: Weak ordering relationship for exponentiation of a fixed real base in [0, 1] to integer exponents. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
Assertion
Ref Expression
leexp2r (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))

Proof of Theorem leexp2r
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . . . . . . 8 (𝑗 = 𝑀 → (𝐴𝑗) = (𝐴𝑀))
21breq1d 5153 . . . . . . 7 (𝑗 = 𝑀 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑀) ≤ (𝐴𝑀)))
32imbi2d 340 . . . . . 6 (𝑗 = 𝑀 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ≤ (𝐴𝑀))))
4 oveq2 7439 . . . . . . . 8 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
54breq1d 5153 . . . . . . 7 (𝑗 = 𝑘 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑘) ≤ (𝐴𝑀)))
65imbi2d 340 . . . . . 6 (𝑗 = 𝑘 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑘) ≤ (𝐴𝑀))))
7 oveq2 7439 . . . . . . . 8 (𝑗 = (𝑘 + 1) → (𝐴𝑗) = (𝐴↑(𝑘 + 1)))
87breq1d 5153 . . . . . . 7 (𝑗 = (𝑘 + 1) → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
98imbi2d 340 . . . . . 6 (𝑗 = (𝑘 + 1) → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀))))
10 oveq2 7439 . . . . . . . 8 (𝑗 = 𝑁 → (𝐴𝑗) = (𝐴𝑁))
1110breq1d 5153 . . . . . . 7 (𝑗 = 𝑁 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑁) ≤ (𝐴𝑀)))
1211imbi2d 340 . . . . . 6 (𝑗 = 𝑁 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))))
13 reexpcl 14119 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴𝑀) ∈ ℝ)
1413adantr 480 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ∈ ℝ)
1514leidd 11829 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ≤ (𝐴𝑀))
16 simprll 779 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ∈ ℝ)
17 1red 11262 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 1 ∈ ℝ)
18 simprlr 780 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑀 ∈ ℕ0)
19 simpl 482 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑘 ∈ (ℤ𝑀))
20 eluznn0 12959 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ ℕ0)
2118, 19, 20syl2anc 584 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑘 ∈ ℕ0)
22 reexpcl 14119 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ)
2316, 21, 22syl2anc 584 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) ∈ ℝ)
24 simprrl 781 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 0 ≤ 𝐴)
25 expge0 14139 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴𝑘))
2616, 21, 24, 25syl3anc 1373 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 0 ≤ (𝐴𝑘))
27 simprrr 782 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ≤ 1)
2816, 17, 23, 26, 27lemul2ad 12208 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) · 𝐴) ≤ ((𝐴𝑘) · 1))
2916recnd 11289 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
30 expp1 14109 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴𝑘) · 𝐴))
3129, 21, 30syl2anc 584 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) = ((𝐴𝑘) · 𝐴))
3223recnd 11289 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) ∈ ℂ)
3332mulridd 11278 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) · 1) = (𝐴𝑘))
3433eqcomd 2743 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) = ((𝐴𝑘) · 1))
3528, 31, 343brtr4d 5175 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘))
36 peano2nn0 12566 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
3721, 36syl 17 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝑘 + 1) ∈ ℕ0)
38 reexpcl 14119 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ ℕ0) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
3916, 37, 38syl2anc 584 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
4013ad2antrl 728 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑀) ∈ ℝ)
41 letr 11355 . . . . . . . . . 10 (((𝐴↑(𝑘 + 1)) ∈ ℝ ∧ (𝐴𝑘) ∈ ℝ ∧ (𝐴𝑀) ∈ ℝ) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ (𝐴𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4239, 23, 40, 41syl3anc 1373 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ (𝐴𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4335, 42mpand 695 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) ≤ (𝐴𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4443ex 412 . . . . . . 7 (𝑘 ∈ (ℤ𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → ((𝐴𝑘) ≤ (𝐴𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀))))
4544a2d 29 . . . . . 6 (𝑘 ∈ (ℤ𝑀) → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑘) ≤ (𝐴𝑀)) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀))))
463, 6, 9, 12, 15, 45uzind4i 12952 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀)))
4746expd 415 . . . 4 (𝑁 ∈ (ℤ𝑀) → ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((0 ≤ 𝐴𝐴 ≤ 1) → (𝐴𝑁) ≤ (𝐴𝑀))))
4847com12 32 . . 3 ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑁 ∈ (ℤ𝑀) → ((0 ≤ 𝐴𝐴 ≤ 1) → (𝐴𝑁) ≤ (𝐴𝑀))))
49483impia 1118 . 2 ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) → ((0 ≤ 𝐴𝐴 ≤ 1) → (𝐴𝑁) ≤ (𝐴𝑀)))
5049imp 406 1 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  cle 11296  0cn0 12526  cuz 12878  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-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-exp 14103
This theorem is referenced by:  exple1  14216  leexp2rd  14294
  Copyright terms: Public domain W3C validator