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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 12399 | . . . 4 ⊢ 1 < 2 | |
2 | nncn 12236 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℂ) | |
3 | 1cnd 11225 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 1 ∈ ℂ) | |
4 | 2, 3 | nncand 11592 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − (𝐼 − 1)) = 1) |
5 | 4 | oveq2d 7430 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − (𝐼 − 1))) = (2↑1)) |
6 | 2cn 12303 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ∈ ℂ) |
8 | 2ne0 12332 | . . . . . . 7 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ≠ 0) |
10 | nnz 12595 | . . . . . . 7 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ) | |
11 | peano2zm 12621 | . . . . . . 7 ⊢ (𝐼 ∈ ℤ → (𝐼 − 1) ∈ ℤ) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − 1) ∈ ℤ) |
13 | 7, 9, 12, 10 | expsubd 14139 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − (𝐼 − 1))) = ((2↑𝐼) / (2↑(𝐼 − 1)))) |
14 | exp1 14050 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑1) = 2) | |
15 | 6, 14 | mp1i 13 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑1) = 2) |
16 | 5, 13, 15 | 3eqtr3d 2775 | . . . 4 ⊢ (𝐼 ∈ ℕ → ((2↑𝐼) / (2↑(𝐼 − 1))) = 2) |
17 | 1, 16 | breqtrrid 5180 | . . 3 ⊢ (𝐼 ∈ ℕ → 1 < ((2↑𝐼) / (2↑(𝐼 − 1)))) |
18 | 2nn 12301 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 2 ∈ ℕ) |
20 | nnm1nn0 12529 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → (𝐼 − 1) ∈ ℕ0) | |
21 | 19, 20 | nnexpcld 14225 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℕ) |
22 | 21 | nnrpd 13032 | . . . 4 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℝ+) |
23 | 2z 12610 | . . . . . 6 ⊢ 2 ∈ ℤ | |
24 | nnnn0 12495 | . . . . . 6 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℕ0) | |
25 | zexpcl 14059 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝐼 ∈ ℕ0) → (2↑𝐼) ∈ ℤ) | |
26 | 23, 24, 25 | sylancr 586 | . . . . 5 ⊢ (𝐼 ∈ ℕ → (2↑𝐼) ∈ ℤ) |
27 | 26 | zred 12682 | . . . 4 ⊢ (𝐼 ∈ ℕ → (2↑𝐼) ∈ ℝ) |
28 | divgt1b 47494 | . . . 4 ⊢ (((2↑(𝐼 − 1)) ∈ ℝ+ ∧ (2↑𝐼) ∈ ℝ) → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ 1 < ((2↑𝐼) / (2↑(𝐼 − 1))))) | |
29 | 22, 27, 28 | syl2anc 583 | . . 3 ⊢ (𝐼 ∈ ℕ → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ 1 < ((2↑𝐼) / (2↑(𝐼 − 1))))) |
30 | 17, 29 | mpbird 257 | . 2 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) < (2↑𝐼)) |
31 | 21 | nnzd 12601 | . . 3 ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ∈ ℤ) |
32 | zltlem1 12631 | . . 3 ⊢ (((2↑(𝐼 − 1)) ∈ ℤ ∧ (2↑𝐼) ∈ ℤ) → ((2↑(𝐼 − 1)) < (2↑𝐼) ↔ (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1))) | |
33 | 31, 26, 32 | syl2anc 583 | . 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 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 (class class class)co 7414 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 < clt 11264 ≤ cle 11265 − cmin 11460 / cdiv 11887 ℕcn 12228 2c2 12283 ℕ0cn0 12488 ℤcz 12574 ℝ+crp 12992 ↑cexp 14044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-seq 13985 df-exp 14045 |
This theorem is referenced by: logbpw2m1 47553 |
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