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Mirrors > Home > ILE Home > Th. List > geo2sum2 | 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 9337 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | fzoval 10214 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1))) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) = (0...(𝑁 − 1))) |
4 | 3 | sumeq1d 11509 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)) |
5 | 2cn 9053 | . . . 4 ⊢ 2 ∈ ℂ | |
6 | 5 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) |
7 | 1ap2 9189 | . . . . 5 ⊢ 1 # 2 | |
8 | ax-1cn 7965 | . . . . . 6 ⊢ 1 ∈ ℂ | |
9 | apsym 8625 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ 2 ∈ ℂ) → (1 # 2 ↔ 2 # 1)) | |
10 | 8, 5, 9 | mp2an 426 | . . . . 5 ⊢ (1 # 2 ↔ 2 # 1) |
11 | 7, 10 | mpbi 145 | . . . 4 ⊢ 2 # 1 |
12 | 11 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 # 1) |
13 | id 19 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
14 | 6, 12, 13 | geoserap 11650 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
15 | 6, 13 | expcld 10744 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℂ) |
16 | 8 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) |
17 | 15, 16 | subcld 8330 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) − 1) ∈ ℂ) |
18 | 1ap0 8609 | . . . . 5 ⊢ 1 # 0 | |
19 | 18 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 1 # 0) |
20 | 17, 16, 19 | div2negapd 8824 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1)) |
21 | 15, 16 | negsubdi2d 8346 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -((2↑𝑁) − 1) = (1 − (2↑𝑁))) |
22 | 2m1e1 9100 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
23 | 22 | negeqi 8213 | . . . . . 6 ⊢ -(2 − 1) = -1 |
24 | 5, 8 | negsubdi2i 8305 | . . . . . 6 ⊢ -(2 − 1) = (1 − 2) |
25 | 23, 24 | eqtr3i 2216 | . . . . 5 ⊢ -1 = (1 − 2) |
26 | 25 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -1 = (1 − 2)) |
27 | 21, 26 | oveq12d 5936 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2))) |
28 | 17 | div1d 8799 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1)) |
29 | 20, 27, 28 | 3eqtr3d 2234 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1)) |
30 | 4, 14, 29 | 3eqtrd 2230 | 1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = ((2↑𝑁) − 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2164 class class class wbr 4029 (class class class)co 5918 ℂcc 7870 0cc0 7872 1c1 7873 − cmin 8190 -cneg 8191 # cap 8600 / cdiv 8691 2c2 9033 ℕ0cn0 9240 ℤcz 9317 ...cfz 10074 ..^cfzo 10208 ↑cexp 10609 Σcsu 11496 |
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 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 |
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 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497 |
This theorem is referenced by: (None) |
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