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Mirrors > Home > MPE Home > Th. List > Mathboxes > pw2m1lepw2m1 | Structured version Visualization version GIF version |
Description: 2 to the power of a positive integer decreased by 1 is less than or equal to 2 to the power of the integer minus 1. (Contributed by AV, 30-May-2020.) |
Ref | Expression |
---|---|
pw2m1lepw2m1 | ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2 11807 | . . . 4 ⊢ 1 < 2 | |
2 | nncn 11645 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℂ) | |
3 | 1cnd 10635 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 1 ∈ ℂ) | |
4 | 2, 3 | nncand 11001 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − (𝐼 − 1)) = 1) |
5 | 4 | oveq2d 7171 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − (𝐼 − 1))) = (2↑1)) |
6 | 2cn 11711 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ∈ ℂ) |
8 | 2ne0 11740 | . . . . . . 7 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ≠ 0) |
10 | nnz 12003 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ) | |
11 | peano2zm 12024 | . . . . . . 7 ⊢ (𝐼 ∈ ℤ → (𝐼 − 1) ∈ ℤ) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − 1) ∈ ℤ) |
13 | 7, 9, 12, 10 | expsubd 13520 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − (𝐼 − 1))) = ((2↑𝐼) / (2↑(𝐼 − 1)))) |
14 | exp1 13434 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑1) = 2) | |
15 | 6, 14 | mp1i 13 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑1) = 2) |
16 | 5, 13, 15 | 3eqtr3d 2864 | . . . 4 ⊢ (𝐼 ∈ ℕ → ((2↑𝐼) / (2↑(𝐼 − 1))) = 2) |
17 | 1, 16 | breqtrrid 5103 | . . 3 ⊢ (𝐼 ∈ ℕ → 1 < ((2↑𝐼) / (2↑(𝐼 − 1)))) |
18 | 2nn 11709 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ∈ ℕ) |
20 | nnm1nn0 11937 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − 1) ∈ ℕ0) | |
21 | 19, 20 | nnexpcld 13605 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℕ) |
22 | 21 | nnrpd 12428 | . . . 4 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℝ+) |
23 | 2z 12013 | . . . . . 6 ⊢ 2 ∈ ℤ | |
24 | nnnn0 11903 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℕ0) | |
25 | zexpcl 13443 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝐼 ∈ ℕ0) → (2↑𝐼) ∈ ℤ) | |
26 | 23, 24, 25 | sylancr 589 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑𝐼) ∈ ℤ) |
27 | 26 | zred 12086 | . . . 4 ⊢ (𝐼 ∈ ℕ → (2↑𝐼) ∈ ℝ) |
28 | divgt1b 44567 | . . . 4 ⊢ (((2↑(𝐼 − 1)) ∈ ℝ+ ∧ (2↑𝐼) ∈ ℝ) → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ 1 < ((2↑𝐼) / (2↑(𝐼 − 1))))) | |
29 | 22, 27, 28 | syl2anc 586 | . . 3 ⊢ (𝐼 ∈ ℕ → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ 1 < ((2↑𝐼) / (2↑(𝐼 − 1))))) |
30 | 17, 29 | mpbird 259 | . 2 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) < (2↑𝐼)) |
31 | 21 | nnzd 12085 | . . 3 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℤ) |
32 | zltlem1 12034 | . . 3 ⊢ (((2↑(𝐼 − 1)) ∈ ℤ ∧ (2↑𝐼) ∈ ℤ) → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1))) | |
33 | 31, 26, 32 | syl2anc 586 | . 2 ⊢ (𝐼 ∈ ℕ → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1))) |
34 | 30, 33 | mpbid 234 | 1 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 < clt 10674 ≤ cle 10675 − cmin 10869 / cdiv 11296 ℕcn 11637 2c2 11691 ℕ0cn0 11896 ℤcz 11980 ℝ+crp 12388 ↑cexp 13428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-seq 13369 df-exp 13429 |
This theorem is referenced by: logbpw2m1 44626 |
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