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Theorem leexp2r 14139
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 7417 . . . . . . . 8 (𝑗 = 𝑀 → (𝐴𝑗) = (𝐴𝑀))
21breq1d 5159 . . . . . . 7 (𝑗 = 𝑀 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑀) ≤ (𝐴𝑀)))
32imbi2d 341 . . . . . 6 (𝑗 = 𝑀 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ≤ (𝐴𝑀))))
4 oveq2 7417 . . . . . . . 8 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
54breq1d 5159 . . . . . . 7 (𝑗 = 𝑘 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑘) ≤ (𝐴𝑀)))
65imbi2d 341 . . . . . 6 (𝑗 = 𝑘 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑘) ≤ (𝐴𝑀))))
7 oveq2 7417 . . . . . . . 8 (𝑗 = (𝑘 + 1) → (𝐴𝑗) = (𝐴↑(𝑘 + 1)))
87breq1d 5159 . . . . . . 7 (𝑗 = (𝑘 + 1) → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
98imbi2d 341 . . . . . 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 7417 . . . . . . . 8 (𝑗 = 𝑁 → (𝐴𝑗) = (𝐴𝑁))
1110breq1d 5159 . . . . . . 7 (𝑗 = 𝑁 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑁) ≤ (𝐴𝑀)))
1211imbi2d 341 . . . . . 6 (𝑗 = 𝑁 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))))
13 reexpcl 14044 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴𝑀) ∈ ℝ)
1413adantr 482 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ∈ ℝ)
1514leidd 11780 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ≤ (𝐴𝑀))
16 simprll 778 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ∈ ℝ)
17 1red 11215 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 1 ∈ ℝ)
18 simprlr 779 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑀 ∈ ℕ0)
19 simpl 484 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑘 ∈ (ℤ𝑀))
20 eluznn0 12901 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ ℕ0)
2118, 19, 20syl2anc 585 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑘 ∈ ℕ0)
22 reexpcl 14044 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ)
2316, 21, 22syl2anc 585 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) ∈ ℝ)
24 simprrl 780 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 0 ≤ 𝐴)
25 expge0 14064 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴𝑘))
2616, 21, 24, 25syl3anc 1372 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 0 ≤ (𝐴𝑘))
27 simprrr 781 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ≤ 1)
2816, 17, 23, 26, 27lemul2ad 12154 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) · 𝐴) ≤ ((𝐴𝑘) · 1))
2916recnd 11242 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
30 expp1 14034 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴𝑘) · 𝐴))
3129, 21, 30syl2anc 585 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) = ((𝐴𝑘) · 𝐴))
3223recnd 11242 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) ∈ ℂ)
3332mulridd 11231 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) · 1) = (𝐴𝑘))
3433eqcomd 2739 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) = ((𝐴𝑘) · 1))
3528, 31, 343brtr4d 5181 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘))
36 peano2nn0 12512 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
3721, 36syl 17 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝑘 + 1) ∈ ℕ0)
38 reexpcl 14044 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ ℕ0) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
3916, 37, 38syl2anc 585 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
4013ad2antrl 727 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑀) ∈ ℝ)
41 letr 11308 . . . . . . . . . 10 (((𝐴↑(𝑘 + 1)) ∈ ℝ ∧ (𝐴𝑘) ∈ ℝ ∧ (𝐴𝑀) ∈ ℝ) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ (𝐴𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4239, 23, 40, 41syl3anc 1372 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ (𝐴𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4335, 42mpand 694 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) ≤ (𝐴𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4443ex 414 . . . . . . 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 12894 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀)))
4746expd 417 . . . 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 408 1 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107   class class class wbr 5149  cfv 6544  (class class class)co 7409  cc 11108  cr 11109  0cc0 11110  1c1 11111   + caddc 11113   · cmul 11115  cle 11249  0cn0 12472  cuz 12822  cexp 14027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-seq 13967  df-exp 14028
This theorem is referenced by:  exple1  14141  leexp2rd  14218
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