![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11603 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | fzoval 12680 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) = (0...(𝑁 − 1))) |
4 | 3 | sumeq1d 14640 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)) |
5 | 2cn 11294 | . . . 4 ⊢ 2 ∈ ℂ | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) |
7 | 1ne2 11443 | . . . . 5 ⊢ 1 ≠ 2 | |
8 | 7 | necomi 2997 | . . . 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 14807 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
12 | 6, 10 | expcld 13216 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℂ) |
13 | ax-1cn 10197 | . . . . . 6 ⊢ 1 ∈ ℂ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) |
15 | 12, 14 | subcld 10595 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) − 1) ∈ ℂ) |
16 | ax-1ne0 10208 | . . . . 5 ⊢ 1 ≠ 0 | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 1 ≠ 0) |
18 | 15, 14, 17 | div2negd 11019 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1)) |
19 | 12, 14 | negsubdi2d 10611 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -((2↑𝑁) − 1) = (1 − (2↑𝑁))) |
20 | 2m1e1 11338 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
21 | 20 | negeqi 10477 | . . . . . 6 ⊢ -(2 − 1) = -1 |
22 | 5, 13 | negsubdi2i 10570 | . . . . . 6 ⊢ -(2 − 1) = (1 − 2) |
23 | 21, 22 | eqtr3i 2795 | . . . . 5 ⊢ -1 = (1 − 2) |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -1 = (1 − 2)) |
25 | 19, 24 | oveq12d 6812 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2))) |
26 | 15 | div1d 10996 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1)) |
27 | 18, 25, 26 | 3eqtr3d 2813 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1)) |
28 | 4, 11, 27 | 3eqtrd 2809 | 1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = ((2↑𝑁) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 (class class class)co 6794 ℂcc 10137 0cc0 10139 1c1 10140 − cmin 10469 -cneg 10470 / cdiv 10887 2c2 11273 ℕ0cn0 11495 ℤcz 11580 ...cfz 12534 ..^cfzo 12674 ↑cexp 13068 Σcsu 14625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-inf2 8703 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 ax-pre-sup 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-isom 6041 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-oadd 7718 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-sup 8505 df-oi 8572 df-card 8966 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-div 10888 df-nn 11224 df-2 11282 df-3 11283 df-n0 11496 df-z 11581 df-uz 11890 df-rp 12037 df-fz 12535 df-fzo 12675 df-seq 13010 df-exp 13069 df-hash 13323 df-cj 14048 df-re 14049 df-im 14050 df-sqrt 14184 df-abs 14185 df-clim 14428 df-sum 14626 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |