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Theorem leexp2r 14210
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 7438 . . . . . . . 8 (𝑗 = 𝑀 → (𝐴𝑗) = (𝐴𝑀))
21breq1d 5157 . . . . . . 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 7438 . . . . . . . 8 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
54breq1d 5157 . . . . . . 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 7438 . . . . . . . 8 (𝑗 = (𝑘 + 1) → (𝐴𝑗) = (𝐴↑(𝑘 + 1)))
87breq1d 5157 . . . . . . 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 7438 . . . . . . . 8 (𝑗 = 𝑁 → (𝐴𝑗) = (𝐴𝑁))
1110breq1d 5157 . . . . . . 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 14115 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴𝑀) ∈ ℝ)
1413adantr 480 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ∈ ℝ)
1514leidd 11826 . . . . . 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 11259 . . . . . . . . . . 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 12956 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ ℕ0)
2118, 19, 20syl2anc 584 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑘 ∈ ℕ0)
22 reexpcl 14115 . . . . . . . . . . . 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 14135 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴𝑘))
2616, 21, 24, 25syl3anc 1370 . . . . . . . . . . 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 12205 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) · 𝐴) ≤ ((𝐴𝑘) · 1))
2916recnd 11286 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
30 expp1 14105 . . . . . . . . . . 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 11286 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) ∈ ℂ)
3332mulridd 11275 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) · 1) = (𝐴𝑘))
3433eqcomd 2740 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) = ((𝐴𝑘) · 1))
3528, 31, 343brtr4d 5179 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘))
36 peano2nn0 12563 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
3721, 36syl 17 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝑘 + 1) ∈ ℕ0)
38 reexpcl 14115 . . . . . . . . . . 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 11352 . . . . . . . . . 10 (((𝐴↑(𝑘 + 1)) ∈ ℝ ∧ (𝐴𝑘) ∈ ℝ ∧ (𝐴𝑀) ∈ ℝ) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ (𝐴𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4239, 23, 40, 41syl3anc 1370 . . . . . . . . 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 12949 . . . . 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 1116 . 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 1086   = wceq 1536  wcel 2105   class class class wbr 5147  cfv 6562  (class class class)co 7430  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157  cle 11293  0cn0 12523  cuz 12875  cexp 14098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-seq 14039  df-exp 14099
This theorem is referenced by:  exple1  14212  leexp2rd  14290
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