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Mirrors > Home > MPE Home > Th. List > geoserg | Structured version Visualization version GIF version |
Description: The value of the finite geometric series 𝐴↑𝑀 + 𝐴↑(𝑀 + 1) +... + 𝐴↑(𝑁 − 1). (Contributed by Mario Carneiro, 2-May-2016.) |
Ref | Expression |
---|---|
geoserg.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
geoserg.2 | ⊢ (𝜑 → 𝐴 ≠ 1) |
geoserg.3 | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
geoserg.4 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
Ref | Expression |
---|---|
geoserg | ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) = (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzofi 14011 | . . . . . 6 ⊢ (𝑀..^𝑁) ∈ Fin | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin) |
3 | ax-1cn 11210 | . . . . . 6 ⊢ 1 ∈ ℂ | |
4 | geoserg.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
5 | subcl 11504 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 587 | . . . . 5 ⊢ (𝜑 → (1 − 𝐴) ∈ ℂ) |
7 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ) |
8 | geoserg.3 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
9 | elfzouz 13699 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
10 | eluznn0 12956 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) | |
11 | 8, 9, 10 | syl2an 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℕ0) |
12 | 7, 11 | expcld 14182 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐴↑𝑘) ∈ ℂ) |
13 | 2, 6, 12 | fsummulc1 15817 | . . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) · (1 − 𝐴)) = Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) · (1 − 𝐴))) |
14 | 3 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ) |
15 | 12, 14, 7 | subdid 11716 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · (1 − 𝐴)) = (((𝐴↑𝑘) · 1) − ((𝐴↑𝑘) · 𝐴))) |
16 | 12 | mulridd 11275 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘)) |
17 | 7, 11 | expp1d 14183 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴)) |
18 | 17 | eqcomd 2740 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · 𝐴) = (𝐴↑(𝑘 + 1))) |
19 | 16, 18 | oveq12d 7448 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (((𝐴↑𝑘) · 1) − ((𝐴↑𝑘) · 𝐴)) = ((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
20 | 15, 19 | eqtrd 2774 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · (1 − 𝐴)) = ((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
21 | 20 | sumeq2dv 15734 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) · (1 − 𝐴)) = Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
22 | oveq2 7438 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘)) | |
23 | oveq2 7438 | . . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1))) | |
24 | oveq2 7438 | . . . . 5 ⊢ (𝑗 = 𝑀 → (𝐴↑𝑗) = (𝐴↑𝑀)) | |
25 | oveq2 7438 | . . . . 5 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁)) | |
26 | geoserg.4 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
27 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
28 | elfzuz 13556 | . . . . . . 7 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ (ℤ≥‘𝑀)) | |
29 | eluznn0 12956 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ ℕ0) | |
30 | 8, 28, 29 | syl2an 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℕ0) |
31 | 27, 30 | expcld 14182 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐴↑𝑗) ∈ ℂ) |
32 | 22, 23, 24, 25, 26, 31 | telfsumo 15834 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) − (𝐴↑(𝑘 + 1))) = ((𝐴↑𝑀) − (𝐴↑𝑁))) |
33 | 13, 21, 32 | 3eqtrrd 2779 | . . 3 ⊢ (𝜑 → ((𝐴↑𝑀) − (𝐴↑𝑁)) = (Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) · (1 − 𝐴))) |
34 | 4, 8 | expcld 14182 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℂ) |
35 | eluznn0 12956 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℕ0) | |
36 | 8, 26, 35 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
37 | 4, 36 | expcld 14182 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
38 | 34, 37 | subcld 11617 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑀) − (𝐴↑𝑁)) ∈ ℂ) |
39 | 2, 12 | fsumcl 15765 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) ∈ ℂ) |
40 | geoserg.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 1) | |
41 | 40 | necomd 2993 | . . . . 5 ⊢ (𝜑 → 1 ≠ 𝐴) |
42 | subeq0 11532 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴)) | |
43 | 3, 4, 42 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → ((1 − 𝐴) = 0 ↔ 1 = 𝐴)) |
44 | 43 | necon3bid 2982 | . . . . 5 ⊢ (𝜑 → ((1 − 𝐴) ≠ 0 ↔ 1 ≠ 𝐴)) |
45 | 41, 44 | mpbird 257 | . . . 4 ⊢ (𝜑 → (1 − 𝐴) ≠ 0) |
46 | 38, 39, 6, 45 | divmul3d 12074 | . . 3 ⊢ (𝜑 → ((((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴)) = Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) ↔ ((𝐴↑𝑀) − (𝐴↑𝑁)) = (Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) · (1 − 𝐴)))) |
47 | 33, 46 | mpbird 257 | . 2 ⊢ (𝜑 → (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴)) = Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘)) |
48 | 47 | eqcomd 2740 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) = (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 − cmin 11489 / cdiv 11917 ℕ0cn0 12523 ℤ≥cuz 12875 ...cfz 13543 ..^cfzo 13690 ↑cexp 14098 Σcsu 15718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 |
This theorem is referenced by: geoser 15899 rplogsumlem2 27543 rpvmasumlem 27545 dchrisum0flblem1 27566 |
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