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Mirrors > Home > MPE Home > Th. List > leexp2a | Structured version Visualization version GIF version |
Description: Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.) |
Ref | Expression |
---|---|
leexp2a | ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ≤ (𝐴↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1138 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℝ) | |
2 | 0red 10801 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 0 ∈ ℝ) | |
3 | 1red 10799 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 1 ∈ ℝ) | |
4 | 0lt1 11319 | . . . . . . . . 9 ⊢ 0 < 1 | |
5 | 4 | a1i 11 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 0 < 1) |
6 | simp2 1139 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 1 ≤ 𝐴) | |
7 | 2, 3, 1, 5, 6 | ltletrd 10957 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 0 < 𝐴) |
8 | 1, 7 | elrpd 12590 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℝ+) |
9 | eluzel2 12408 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
10 | 9 | 3ad2ant3 1137 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ) |
11 | rpexpcl 13619 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑀 ∈ ℤ) → (𝐴↑𝑀) ∈ ℝ+) | |
12 | 8, 10, 11 | syl2anc 587 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ∈ ℝ+) |
13 | 12 | rpred 12593 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ∈ ℝ) |
14 | 13 | recnd 10826 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ∈ ℂ) |
15 | 14 | mulid2d 10816 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (1 · (𝐴↑𝑀)) = (𝐴↑𝑀)) |
16 | uznn0sub 12438 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) | |
17 | 16 | 3ad2ant3 1137 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑁 − 𝑀) ∈ ℕ0) |
18 | expge1 13637 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑁 − 𝑀) ∈ ℕ0 ∧ 1 ≤ 𝐴) → 1 ≤ (𝐴↑(𝑁 − 𝑀))) | |
19 | 1, 17, 6, 18 | syl3anc 1373 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 1 ≤ (𝐴↑(𝑁 − 𝑀))) |
20 | 1 | recnd 10826 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℂ) |
21 | 7 | gt0ne0d 11361 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝐴 ≠ 0) |
22 | eluzelz 12413 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
23 | 22 | 3ad2ant3 1137 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ) |
24 | expsub 13648 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝐴↑(𝑁 − 𝑀)) = ((𝐴↑𝑁) / (𝐴↑𝑀))) | |
25 | 20, 21, 23, 10, 24 | syl22anc 839 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑(𝑁 − 𝑀)) = ((𝐴↑𝑁) / (𝐴↑𝑀))) |
26 | 19, 25 | breqtrd 5065 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 1 ≤ ((𝐴↑𝑁) / (𝐴↑𝑀))) |
27 | rpexpcl 13619 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) | |
28 | 8, 23, 27 | syl2anc 587 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑁) ∈ ℝ+) |
29 | 28 | rpred 12593 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑁) ∈ ℝ) |
30 | 3, 29, 12 | lemuldivd 12642 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((1 · (𝐴↑𝑀)) ≤ (𝐴↑𝑁) ↔ 1 ≤ ((𝐴↑𝑁) / (𝐴↑𝑀)))) |
31 | 26, 30 | mpbird 260 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (1 · (𝐴↑𝑀)) ≤ (𝐴↑𝑁)) |
32 | 15, 31 | eqbrtrrd 5063 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ≤ (𝐴↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 1c1 10695 · cmul 10699 < clt 10832 ≤ cle 10833 − cmin 11027 / cdiv 11454 ℕ0cn0 12055 ℤcz 12141 ℤ≥cuz 12403 ℝ+crp 12551 ↑cexp 13600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-seq 13540 df-exp 13601 |
This theorem is referenced by: expnlbnd2 13766 digit1 13769 leexp2ad 13788 faclbnd4lem1 13824 climcndslem1 15376 climcndslem2 15377 ef01bndlem 15708 aaliou3lem2 25190 ackval42 45658 |
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