![]() |
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 12613 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | fzoval 13665 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) = (0...(𝑁 − 1))) |
4 | 3 | sumeq1d 15679 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)) |
5 | 2cn 12317 | . . . 4 ⊢ 2 ∈ ℂ | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) |
7 | 1ne2 12450 | . . . . 5 ⊢ 1 ≠ 2 | |
8 | 7 | necomi 2985 | . . . 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 15845 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
12 | 6, 10 | expcld 14142 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℂ) |
13 | ax-1cn 11196 | . . . . . 6 ⊢ 1 ∈ ℂ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) |
15 | 12, 14 | subcld 11601 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) − 1) ∈ ℂ) |
16 | ax-1ne0 11207 | . . . . 5 ⊢ 1 ≠ 0 | |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 1 ≠ 0) |
18 | 15, 14, 17 | div2negd 12035 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1)) |
19 | 12, 14 | negsubdi2d 11617 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -((2↑𝑁) − 1) = (1 − (2↑𝑁))) |
20 | 2m1e1 12368 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
21 | 20 | negeqi 11483 | . . . . . 6 ⊢ -(2 − 1) = -1 |
22 | 5, 13 | negsubdi2i 11576 | . . . . . 6 ⊢ -(2 − 1) = (1 − 2) |
23 | 21, 22 | eqtr3i 2755 | . . . . 5 ⊢ -1 = (1 − 2) |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → -1 = (1 − 2)) |
25 | 19, 24 | oveq12d 7434 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2))) |
26 | 15 | div1d 12012 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1)) |
27 | 18, 25, 26 | 3eqtr3d 2773 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1)) |
28 | 4, 11, 27 | 3eqtrd 2769 | 1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = ((2↑𝑁) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 (class class class)co 7416 ℂcc 11136 0cc0 11138 1c1 11139 − cmin 11474 -cneg 11475 / cdiv 11901 2c2 12297 ℕ0cn0 12502 ℤcz 12588 ...cfz 13516 ..^cfzo 13659 ↑cexp 14058 Σcsu 15664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |