![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9323 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | fzoval 10200 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1))) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) = (0...(𝑁 − 1))) |
4 | 3 | sumeq1d 11483 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)) |
5 | 2cn 9039 | . . . 4 ⊢ 2 ∈ ℂ | |
6 | 5 | a1i 9 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) |
7 | 1ap2 9175 | . . . . 5 ⊢ 1 # 2 | |
8 | ax-1cn 7951 | . . . . . 6 ⊢ 1 ∈ ℂ | |
9 | apsym 8611 | . . . . . 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 11624 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
15 | 6, 13 | expcld 10718 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℂ) |
16 | 8 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) |
17 | 15, 16 | subcld 8316 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) − 1) ∈ ℂ) |
18 | 1ap0 8595 | . . . . 5 ⊢ 1 # 0 | |
19 | 18 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 1 # 0) |
20 | 17, 16, 19 | div2negapd 8810 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1)) |
21 | 15, 16 | negsubdi2d 8332 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -((2↑𝑁) − 1) = (1 − (2↑𝑁))) |
22 | 2m1e1 9086 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
23 | 22 | negeqi 8199 | . . . . . 6 ⊢ -(2 − 1) = -1 |
24 | 5, 8 | negsubdi2i 8291 | . . . . . 6 ⊢ -(2 − 1) = (1 − 2) |
25 | 23, 24 | eqtr3i 2212 | . . . . 5 ⊢ -1 = (1 − 2) |
26 | 25 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -1 = (1 − 2)) |
27 | 21, 26 | oveq12d 5924 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2))) |
28 | 17 | div1d 8785 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1)) |
29 | 20, 27, 28 | 3eqtr3d 2230 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1)) |
30 | 4, 14, 29 | 3eqtrd 2226 | 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 2160 class class class wbr 4025 (class class class)co 5906 ℂcc 7856 0cc0 7858 1c1 7859 − cmin 8176 -cneg 8177 # cap 8586 / cdiv 8677 2c2 9019 ℕ0cn0 9226 ℤcz 9303 ...cfz 10060 ..^cfzo 10194 ↑cexp 10583 Σcsu 11470 |
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 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 ax-arch 7977 ax-caucvg 7978 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-id 4318 df-po 4321 df-iso 4322 df-iord 4391 df-on 4393 df-ilim 4394 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-isom 5251 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-recs 6345 df-irdg 6410 df-frec 6431 df-1o 6456 df-oadd 6460 df-er 6574 df-en 6782 df-dom 6783 df-fin 6784 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-2 9027 df-3 9028 df-4 9029 df-n0 9227 df-z 9304 df-uz 9579 df-q 9671 df-rp 9706 df-fz 10061 df-fzo 10195 df-seqfrec 10505 df-exp 10584 df-ihash 10821 df-cj 10960 df-re 10961 df-im 10962 df-rsqrt 11116 df-abs 11117 df-clim 11396 df-sumdc 11471 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |