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Theorem leexp2r 12866
 Description: Weak ordering relationship for exponentiation. (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 6618 . . . . . . . 8 (𝑗 = 𝑀 → (𝐴𝑗) = (𝐴𝑀))
21breq1d 4628 . . . . . . 7 (𝑗 = 𝑀 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑀) ≤ (𝐴𝑀)))
32imbi2d 330 . . . . . 6 (𝑗 = 𝑀 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ≤ (𝐴𝑀))))
4 oveq2 6618 . . . . . . . 8 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
54breq1d 4628 . . . . . . 7 (𝑗 = 𝑘 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑘) ≤ (𝐴𝑀)))
65imbi2d 330 . . . . . 6 (𝑗 = 𝑘 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑘) ≤ (𝐴𝑀))))
7 oveq2 6618 . . . . . . . 8 (𝑗 = (𝑘 + 1) → (𝐴𝑗) = (𝐴↑(𝑘 + 1)))
87breq1d 4628 . . . . . . 7 (𝑗 = (𝑘 + 1) → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
98imbi2d 330 . . . . . 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 6618 . . . . . . . 8 (𝑗 = 𝑁 → (𝐴𝑗) = (𝐴𝑁))
1110breq1d 4628 . . . . . . 7 (𝑗 = 𝑁 → ((𝐴𝑗) ≤ (𝐴𝑀) ↔ (𝐴𝑁) ≤ (𝐴𝑀)))
1211imbi2d 330 . . . . . 6 (𝑗 = 𝑁 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑗) ≤ (𝐴𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))))
13 reexpcl 12825 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴𝑀) ∈ ℝ)
1413adantr 481 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ∈ ℝ)
1514leidd 10546 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ≤ (𝐴𝑀))
1615a1i 11 . . . . . 6 (𝑀 ∈ ℤ → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑀) ≤ (𝐴𝑀)))
17 simprll 801 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ∈ ℝ)
18 1red 10007 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 1 ∈ ℝ)
19 simprlr 802 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑀 ∈ ℕ0)
20 simpl 473 . . . . . . . . . . . . 13 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑘 ∈ (ℤ𝑀))
21 eluznn0 11709 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ ℕ0)
2219, 20, 21syl2anc 692 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝑘 ∈ ℕ0)
23 reexpcl 12825 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ)
2417, 22, 23syl2anc 692 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) ∈ ℝ)
25 simprrl 803 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 0 ≤ 𝐴)
26 expge0 12844 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴𝑘))
2717, 22, 25, 26syl3anc 1323 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 0 ≤ (𝐴𝑘))
28 simprrr 804 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ≤ 1)
2917, 18, 24, 27, 28lemul2ad 10916 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) · 𝐴) ≤ ((𝐴𝑘) · 1))
3017recnd 10020 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
31 expp1 12815 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴𝑘) · 𝐴))
3230, 22, 31syl2anc 692 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) = ((𝐴𝑘) · 𝐴))
3324recnd 10020 . . . . . . . . . . . 12 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) ∈ ℂ)
3433mulid1d 10009 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) · 1) = (𝐴𝑘))
3534eqcomd 2627 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑘) = ((𝐴𝑘) · 1))
3629, 32, 353brtr4d 4650 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘))
37 peano2nn0 11285 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
3822, 37syl 17 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝑘 + 1) ∈ ℕ0)
39 reexpcl 12825 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ ℕ0) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
4017, 38, 39syl2anc 692 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
4113ad2antrl 763 . . . . . . . . . 10 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (𝐴𝑀) ∈ ℝ)
42 letr 10083 . . . . . . . . . 10 (((𝐴↑(𝑘 + 1)) ∈ ℝ ∧ (𝐴𝑘) ∈ ℝ ∧ (𝐴𝑀) ∈ ℝ) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ (𝐴𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4340, 24, 41, 42syl3anc 1323 . . . . . . . . 9 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴𝑘) ∧ (𝐴𝑘) ≤ (𝐴𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4436, 43mpand 710 . . . . . . . 8 ((𝑘 ∈ (ℤ𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1))) → ((𝐴𝑘) ≤ (𝐴𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀)))
4544ex 450 . . . . . . 7 (𝑘 ∈ (ℤ𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → ((𝐴𝑘) ≤ (𝐴𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀))))
4645a2d 29 . . . . . 6 (𝑘 ∈ (ℤ𝑀) → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑘) ≤ (𝐴𝑀)) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴𝑀))))
473, 6, 9, 12, 16, 46uzind4 11698 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀)))
4847expd 452 . . . 4 (𝑁 ∈ (ℤ𝑀) → ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((0 ≤ 𝐴𝐴 ≤ 1) → (𝐴𝑁) ≤ (𝐴𝑀))))
4948com12 32 . . 3 ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑁 ∈ (ℤ𝑀) → ((0 ≤ 𝐴𝐴 ≤ 1) → (𝐴𝑁) ≤ (𝐴𝑀))))
50493impia 1258 . 2 ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) → ((0 ≤ 𝐴𝐴 ≤ 1) → (𝐴𝑁) ≤ (𝐴𝑀)))
5150imp 445 1 (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   class class class wbr 4618  ‘cfv 5852  (class class class)co 6610  ℂcc 9886  ℝcr 9887  0cc0 9888  1c1 9889   + caddc 9891   · cmul 9893   ≤ cle 10027  ℕ0cn0 11244  ℤcz 11329  ℤ≥cuz 11639  ↑cexp 12808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-seq 12750  df-exp 12809 This theorem is referenced by:  exple1  12868  leexp2rd  12990
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