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Mirrors > Home > MPE Home > Th. List > geo2sum2 | Structured version Visualization version GIF version |
Description: The value of the finite geometric series 1 + 2 + 4 + 8 +... + 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 7-Sep-2016.) |
Ref | Expression |
---|---|
geo2sum2 | ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = ((2↑𝑁) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 11993 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | fzoval 13027 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) = (0...(𝑁 − 1))) |
4 | 3 | sumeq1d 15046 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)) |
5 | 2cn 11700 | . . . 4 ⊢ 2 ∈ ℂ | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) |
7 | 1ne2 11833 | . . . . 5 ⊢ 1 ≠ 2 | |
8 | 7 | necomi 3067 | . . . 4 ⊢ 2 ≠ 1 |
9 | 8 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ≠ 1) |
10 | id 22 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
11 | 6, 9, 10 | geoser 15210 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
12 | 6, 10 | expcld 13498 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℂ) |
13 | ax-1cn 10583 | . . . . . 6 ⊢ 1 ∈ ℂ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) |
15 | 12, 14 | subcld 10985 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) − 1) ∈ ℂ) |
16 | ax-1ne0 10594 | . . . . 5 ⊢ 1 ≠ 0 | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 1 ≠ 0) |
18 | 15, 14, 17 | div2negd 11419 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1)) |
19 | 12, 14 | negsubdi2d 11001 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -((2↑𝑁) − 1) = (1 − (2↑𝑁))) |
20 | 2m1e1 11751 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
21 | 20 | negeqi 10867 | . . . . . 6 ⊢ -(2 − 1) = -1 |
22 | 5, 13 | negsubdi2i 10960 | . . . . . 6 ⊢ -(2 − 1) = (1 − 2) |
23 | 21, 22 | eqtr3i 2843 | . . . . 5 ⊢ -1 = (1 − 2) |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -1 = (1 − 2)) |
25 | 19, 24 | oveq12d 7163 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2))) |
26 | 15 | div1d 11396 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1)) |
27 | 18, 25, 26 | 3eqtr3d 2861 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1)) |
28 | 4, 11, 27 | 3eqtrd 2857 | 1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = ((2↑𝑁) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 − cmin 10858 -cneg 10859 / cdiv 11285 2c2 11680 ℕ0cn0 11885 ℤcz 11969 ...cfz 12880 ..^cfzo 13021 ↑cexp 13417 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 |
This theorem is referenced by: (None) |
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