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Mirrors > Home > MPE Home > Th. List > geoser | Structured version Visualization version GIF version |
Description: The value of the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Proof shortened by Mario Carneiro, 15-Jun-2014.) |
Ref | Expression |
---|---|
geoser.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
geoser.2 | ⊢ (𝜑 → 𝐴 ≠ 1) |
geoser.3 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
geoser | ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | geoser.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | geoser.2 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 1) | |
3 | 0nn0 11513 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
5 | geoser.3 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | nn0uz 11928 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | 5, 6 | syl6eleq 2860 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
8 | 1, 2, 4, 7 | geoserg 14804 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0..^𝑁)(𝐴↑𝑘) = (((𝐴↑0) − (𝐴↑𝑁)) / (1 − 𝐴))) |
9 | 5 | nn0zd 11686 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
10 | fzoval 12678 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1))) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → (0..^𝑁) = (0...(𝑁 − 1))) |
12 | 11 | sumeq1d 14638 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0..^𝑁)(𝐴↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)) |
13 | 1 | exp0d 13208 | . . . 4 ⊢ (𝜑 → (𝐴↑0) = 1) |
14 | 13 | oveq1d 6810 | . . 3 ⊢ (𝜑 → ((𝐴↑0) − (𝐴↑𝑁)) = (1 − (𝐴↑𝑁))) |
15 | 14 | oveq1d 6810 | . 2 ⊢ (𝜑 → (((𝐴↑0) − (𝐴↑𝑁)) / (1 − 𝐴)) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
16 | 8, 12, 15 | 3eqtr3d 2813 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ‘cfv 6030 (class class class)co 6795 ℂcc 10139 0cc0 10141 1c1 10142 − cmin 10471 / cdiv 10889 ℕ0cn0 11498 ℤcz 11583 ℤ≥cuz 11892 ...cfz 12532 ..^cfzo 12672 ↑cexp 13066 Σcsu 14623 |
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 7099 ax-inf2 8705 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 ax-pre-sup 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 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 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 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 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-om 7216 df-1st 7318 df-2nd 7319 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-oadd 7720 df-er 7899 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-sup 8507 df-oi 8574 df-card 8968 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-div 10890 df-nn 11226 df-2 11284 df-3 11285 df-n0 11499 df-z 11584 df-uz 11893 df-rp 12035 df-fz 12533 df-fzo 12673 df-seq 13008 df-exp 13067 df-hash 13321 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-clim 14426 df-sum 14624 |
This theorem is referenced by: pwm1geoser 14806 geolim 14807 geolim2 14808 geo2sum 14810 geo2sum2 14811 3dvds 15260 3dvdsOLD 15261 1sgm2ppw 25145 mersenne 25172 knoppndvlem14 32852 |
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