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Mirrors > Home > MPE Home > Th. List > Mathboxes > expnegico01 | Structured version Visualization version GIF version |
Description: An integer greater than 1 to the power of a negative integer is in the closed-below, open-above interval between 0 and 1. (Contributed by AV, 24-May-2020.) |
Ref | Expression |
---|---|
expnegico01 | ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵↑𝑁) ∈ (0[,)1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelre 11736 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℝ) | |
2 | 1 | adantr 480 | . . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 𝐵 ∈ ℝ) |
3 | eluz2nn 11764 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℕ) | |
4 | 3 | nnne0d 11103 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ≠ 0) |
5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 𝐵 ≠ 0) |
6 | simpr 476 | . . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
7 | 2, 5, 6 | 3jca 1261 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ∧ 𝑁 ∈ ℤ)) |
8 | 7 | 3adant3 1101 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ∧ 𝑁 ∈ ℤ)) |
9 | reexpclz 12920 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐵↑𝑁) ∈ ℝ) | |
10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵↑𝑁) ∈ ℝ) |
11 | 0red 10079 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 0 ∈ ℝ) | |
12 | 1 | 3ad2ant1 1102 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 𝐵 ∈ ℝ) |
13 | 4 | 3ad2ant1 1102 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 𝐵 ≠ 0) |
14 | simp2 1082 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 𝑁 ∈ ℤ) | |
15 | 12, 13, 14 | reexpclzd 13074 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵↑𝑁) ∈ ℝ) |
16 | 3 | nngt0d 11102 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘2) → 0 < 𝐵) |
17 | 16 | 3ad2ant1 1102 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 0 < 𝐵) |
18 | expgt0 12933 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0 < 𝐵) → 0 < (𝐵↑𝑁)) | |
19 | 12, 14, 17, 18 | syl3anc 1366 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 0 < (𝐵↑𝑁)) |
20 | 11, 15, 19 | ltled 10223 | . 2 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 0 ≤ (𝐵↑𝑁)) |
21 | 0zd 11427 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 0 ∈ ℤ) | |
22 | eluz2gt1 11798 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘2) → 1 < 𝐵) | |
23 | 22 | 3ad2ant1 1102 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 1 < 𝐵) |
24 | simp3 1083 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 𝑁 < 0) | |
25 | ltexp2a 12952 | . . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (1 < 𝐵 ∧ 𝑁 < 0)) → (𝐵↑𝑁) < (𝐵↑0)) | |
26 | 12, 14, 21, 23, 24, 25 | syl32anc 1374 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵↑𝑁) < (𝐵↑0)) |
27 | eluzelcn 11737 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℂ) | |
28 | 27 | exp0d 13042 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘2) → (𝐵↑0) = 1) |
29 | 28 | eqcomd 2657 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘2) → 1 = (𝐵↑0)) |
30 | 29 | 3ad2ant1 1102 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → 1 = (𝐵↑0)) |
31 | 26, 30 | breqtrrd 4713 | . 2 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵↑𝑁) < 1) |
32 | 0re 10078 | . . . 4 ⊢ 0 ∈ ℝ | |
33 | 1re 10077 | . . . . 5 ⊢ 1 ∈ ℝ | |
34 | 33 | rexri 10135 | . . . 4 ⊢ 1 ∈ ℝ* |
35 | 32, 34 | pm3.2i 470 | . . 3 ⊢ (0 ∈ ℝ ∧ 1 ∈ ℝ*) |
36 | elico2 12275 | . . 3 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → ((𝐵↑𝑁) ∈ (0[,)1) ↔ ((𝐵↑𝑁) ∈ ℝ ∧ 0 ≤ (𝐵↑𝑁) ∧ (𝐵↑𝑁) < 1))) | |
37 | 35, 36 | mp1i 13 | . 2 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → ((𝐵↑𝑁) ∈ (0[,)1) ↔ ((𝐵↑𝑁) ∈ ℝ ∧ 0 ≤ (𝐵↑𝑁) ∧ (𝐵↑𝑁) < 1))) |
38 | 10, 20, 31, 37 | mpbir3and 1264 | 1 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵↑𝑁) ∈ (0[,)1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 ℝ*cxr 10111 < clt 10112 ≤ cle 10113 2c2 11108 ℤcz 11415 ℤ≥cuz 11725 [,)cico 12215 ↑cexp 12900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-ico 12219 df-seq 12842 df-exp 12901 |
This theorem is referenced by: digexp 42726 |
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