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Mirrors > Home > ILE Home > Th. List > geoserap | 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.) (Revised by Jim Kingdon, 24-Oct-2022.) |
Ref | Expression |
---|---|
geoser.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
geoser.2 | ⊢ (𝜑 → 𝐴 # 1) |
geoser.3 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
geoserap | ⊢ (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 9249 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
5 | geoser.3 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | nn0uz 9621 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | 5, 6 | eleqtrdi 2286 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
8 | 1, 2, 4, 7 | geosergap 11643 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0..^𝑁)(𝐴↑𝑘) = (((𝐴↑0) − (𝐴↑𝑁)) / (1 − 𝐴))) |
9 | 5 | nn0zd 9431 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
10 | fzoval 10208 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1))) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ (𝜑 → (0..^𝑁) = (0...(𝑁 − 1))) |
12 | 11 | sumeq1d 11503 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0..^𝑁)(𝐴↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)) |
13 | 1 | exp0d 10732 | . . . 4 ⊢ (𝜑 → (𝐴↑0) = 1) |
14 | 13 | oveq1d 5929 | . . 3 ⊢ (𝜑 → ((𝐴↑0) − (𝐴↑𝑁)) = (1 − (𝐴↑𝑁))) |
15 | 14 | oveq1d 5929 | . 2 ⊢ (𝜑 → (((𝐴↑0) − (𝐴↑𝑁)) / (1 − 𝐴)) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
16 | 8, 12, 15 | 3eqtr3d 2234 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2164 class class class wbr 4029 ‘cfv 5250 (class class class)co 5914 ℂcc 7864 0cc0 7866 1c1 7867 − cmin 8184 # cap 8594 / cdiv 8685 ℕ0cn0 9234 ℤcz 9311 ℤ≥cuz 9586 ...cfz 10068 ..^cfzo 10202 ↑cexp 10603 Σcsu 11490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4567 ax-iinf 4618 ax-cnex 7957 ax-resscn 7958 ax-1cn 7959 ax-1re 7960 ax-icn 7961 ax-addcl 7962 ax-addrcl 7963 ax-mulcl 7964 ax-mulrcl 7965 ax-addcom 7966 ax-mulcom 7967 ax-addass 7968 ax-mulass 7969 ax-distr 7970 ax-i2m1 7971 ax-0lt1 7972 ax-1rid 7973 ax-0id 7974 ax-rnegex 7975 ax-precex 7976 ax-cnre 7977 ax-pre-ltirr 7978 ax-pre-ltwlin 7979 ax-pre-lttrn 7980 ax-pre-apti 7981 ax-pre-ltadd 7982 ax-pre-mulgt0 7983 ax-pre-mulext 7984 ax-arch 7985 ax-caucvg 7986 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4621 df-xp 4663 df-rel 4664 df-cnv 4665 df-co 4666 df-dm 4667 df-rn 4668 df-res 4669 df-ima 4670 df-iota 5211 df-fun 5252 df-fn 5253 df-f 5254 df-f1 5255 df-fo 5256 df-f1o 5257 df-fv 5258 df-isom 5259 df-riota 5869 df-ov 5917 df-oprab 5918 df-mpo 5919 df-1st 6188 df-2nd 6189 df-recs 6353 df-irdg 6418 df-frec 6439 df-1o 6464 df-oadd 6468 df-er 6582 df-en 6790 df-dom 6791 df-fin 6792 df-pnf 8050 df-mnf 8051 df-xr 8052 df-ltxr 8053 df-le 8054 df-sub 8186 df-neg 8187 df-reap 8588 df-ap 8595 df-div 8686 df-inn 8977 df-2 9035 df-3 9036 df-4 9037 df-n0 9235 df-z 9312 df-uz 9587 df-q 9679 df-rp 9714 df-fz 10069 df-fzo 10203 df-seqfrec 10513 df-exp 10604 df-ihash 10841 df-cj 10980 df-re 10981 df-im 10982 df-rsqrt 11136 df-abs 11137 df-clim 11416 df-sumdc 11491 |
This theorem is referenced by: pwm1geoserap1 11645 geolim 11648 geolim2 11649 geo2sum 11651 geo2sum2 11652 |
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