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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 12408 | . . . 4 ⊢ 1 < 2 | |
2 | nncn 12245 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℂ) | |
3 | 1cnd 11234 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 1 ∈ ℂ) | |
4 | 2, 3 | nncand 11601 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − (𝐼 − 1)) = 1) |
5 | 4 | oveq2d 7429 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − (𝐼 − 1))) = (2↑1)) |
6 | 2cn 12312 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ∈ ℂ) |
8 | 2ne0 12341 | . . . . . . 7 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ≠ 0) |
10 | nnz 12604 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ) | |
11 | peano2zm 12630 | . . . . . . 7 ⊢ (𝐼 ∈ ℤ → (𝐼 − 1) ∈ ℤ) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − 1) ∈ ℤ) |
13 | 7, 9, 12, 10 | expsubd 14148 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − (𝐼 − 1))) = ((2↑𝐼) / (2↑(𝐼 − 1)))) |
14 | exp1 14059 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑1) = 2) | |
15 | 6, 14 | mp1i 13 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑1) = 2) |
16 | 5, 13, 15 | 3eqtr3d 2773 | . . . 4 ⊢ (𝐼 ∈ ℕ → ((2↑𝐼) / (2↑(𝐼 − 1))) = 2) |
17 | 1, 16 | breqtrrid 5182 | . . 3 ⊢ (𝐼 ∈ ℕ → 1 < ((2↑𝐼) / (2↑(𝐼 − 1)))) |
18 | 2nn 12310 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ∈ ℕ) |
20 | nnm1nn0 12538 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − 1) ∈ ℕ0) | |
21 | 19, 20 | nnexpcld 14234 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℕ) |
22 | 21 | nnrpd 13041 | . . . 4 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℝ+) |
23 | 2z 12619 | . . . . . 6 ⊢ 2 ∈ ℤ | |
24 | nnnn0 12504 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℕ0) | |
25 | zexpcl 14068 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝐼 ∈ ℕ0) → (2↑𝐼) ∈ ℤ) | |
26 | 23, 24, 25 | sylancr 585 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑𝐼) ∈ ℤ) |
27 | 26 | zred 12691 | . . . 4 ⊢ (𝐼 ∈ ℕ → (2↑𝐼) ∈ ℝ) |
28 | divgt1b 47689 | . . . 4 ⊢ (((2↑(𝐼 − 1)) ∈ ℝ+ ∧ (2↑𝐼) ∈ ℝ) → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ 1 < ((2↑𝐼) / (2↑(𝐼 − 1))))) | |
29 | 22, 27, 28 | syl2anc 582 | . . 3 ⊢ (𝐼 ∈ ℕ → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ 1 < ((2↑𝐼) / (2↑(𝐼 − 1))))) |
30 | 17, 29 | mpbird 256 | . 2 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) < (2↑𝐼)) |
31 | 21 | nnzd 12610 | . . 3 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℤ) |
32 | zltlem1 12640 | . . 3 ⊢ (((2↑(𝐼 − 1)) ∈ ℤ ∧ (2↑𝐼) ∈ ℤ) → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1))) | |
33 | 31, 26, 32 | syl2anc 582 | . 2 ⊢ (𝐼 ∈ ℕ → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1))) |
34 | 30, 33 | mpbid 231 | 1 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5144 (class class class)co 7413 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 < clt 11273 ≤ cle 11274 − cmin 11469 / cdiv 11896 ℕcn 12237 2c2 12292 ℕ0cn0 12497 ℤcz 12583 ℝ+crp 13001 ↑cexp 14053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-seq 13994 df-exp 14054 |
This theorem is referenced by: logbpw2m1 47748 |
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