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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 13972 | . . . . . 6 ⊢ (𝑀..^𝑁) ∈ Fin | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin) |
3 | ax-1cn 11197 | . . . . . 6 ⊢ 1 ∈ ℂ | |
4 | geoserg.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
5 | subcl 11490 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 586 | . . . . 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 13669 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
10 | eluznn0 12932 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) | |
11 | 8, 9, 10 | syl2an 595 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℕ0) |
12 | 7, 11 | expcld 14143 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐴↑𝑘) ∈ ℂ) |
13 | 2, 6, 12 | fsummulc1 15764 | . . . 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 11701 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · (1 − 𝐴)) = (((𝐴↑𝑘) · 1) − ((𝐴↑𝑘) · 𝐴))) |
16 | 12 | mulridd 11262 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘)) |
17 | 7, 11 | expp1d 14144 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴)) |
18 | 17 | eqcomd 2734 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · 𝐴) = (𝐴↑(𝑘 + 1))) |
19 | 16, 18 | oveq12d 7438 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (((𝐴↑𝑘) · 1) − ((𝐴↑𝑘) · 𝐴)) = ((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
20 | 15, 19 | eqtrd 2768 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · (1 − 𝐴)) = ((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
21 | 20 | sumeq2dv 15682 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) · (1 − 𝐴)) = Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
22 | oveq2 7428 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘)) | |
23 | oveq2 7428 | . . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1))) | |
24 | oveq2 7428 | . . . . 5 ⊢ (𝑗 = 𝑀 → (𝐴↑𝑗) = (𝐴↑𝑀)) | |
25 | oveq2 7428 | . . . . 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 13530 | . . . . . . 7 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ (ℤ≥‘𝑀)) | |
29 | eluznn0 12932 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ ℕ0) | |
30 | 8, 28, 29 | syl2an 595 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℕ0) |
31 | 27, 30 | expcld 14143 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐴↑𝑗) ∈ ℂ) |
32 | 22, 23, 24, 25, 26, 31 | telfsumo 15781 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) − (𝐴↑(𝑘 + 1))) = ((𝐴↑𝑀) − (𝐴↑𝑁))) |
33 | 13, 21, 32 | 3eqtrrd 2773 | . . 3 ⊢ (𝜑 → ((𝐴↑𝑀) − (𝐴↑𝑁)) = (Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) · (1 − 𝐴))) |
34 | 4, 8 | expcld 14143 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℂ) |
35 | eluznn0 12932 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℕ0) | |
36 | 8, 26, 35 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
37 | 4, 36 | expcld 14143 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
38 | 34, 37 | subcld 11602 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑀) − (𝐴↑𝑁)) ∈ ℂ) |
39 | 2, 12 | fsumcl 15712 | . . . 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 11517 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴)) | |
43 | 3, 4, 42 | sylancr 586 | . . . . . 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 12055 | . . 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 2734 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) = (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ‘cfv 6548 (class class class)co 7420 Fincfn 8964 ℂcc 11137 0cc0 11139 1c1 11140 + caddc 11142 · cmul 11144 − cmin 11475 / cdiv 11902 ℕ0cn0 12503 ℤ≥cuz 12853 ...cfz 13517 ..^cfzo 13660 ↑cexp 14059 Σcsu 15665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 |
This theorem is referenced by: geoser 15846 rplogsumlem2 27431 rpvmasumlem 27433 dchrisum0flblem1 27454 |
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