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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 13345 | . . . . . 6 ⊢ (𝑀..^𝑁) ∈ Fin | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin) |
3 | ax-1cn 10597 | . . . . . 6 ⊢ 1 ∈ ℂ | |
4 | geoserg.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
5 | subcl 10887 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 589 | . . . . 5 ⊢ (𝜑 → (1 − 𝐴) ∈ ℂ) |
7 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ) |
8 | geoserg.3 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
9 | elfzouz 13045 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
10 | eluznn0 12320 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) | |
11 | 8, 9, 10 | syl2an 597 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℕ0) |
12 | 7, 11 | expcld 13513 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐴↑𝑘) ∈ ℂ) |
13 | 2, 6, 12 | fsummulc1 15142 | . . . 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 11098 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · (1 − 𝐴)) = (((𝐴↑𝑘) · 1) − ((𝐴↑𝑘) · 𝐴))) |
16 | 12 | mulid1d 10660 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘)) |
17 | 7, 11 | expp1d 13514 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴)) |
18 | 17 | eqcomd 2829 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · 𝐴) = (𝐴↑(𝑘 + 1))) |
19 | 16, 18 | oveq12d 7176 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (((𝐴↑𝑘) · 1) − ((𝐴↑𝑘) · 𝐴)) = ((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
20 | 15, 19 | eqtrd 2858 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝐴↑𝑘) · (1 − 𝐴)) = ((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
21 | 20 | sumeq2dv 15062 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) · (1 − 𝐴)) = Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) − (𝐴↑(𝑘 + 1)))) |
22 | oveq2 7166 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘)) | |
23 | oveq2 7166 | . . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1))) | |
24 | oveq2 7166 | . . . . 5 ⊢ (𝑗 = 𝑀 → (𝐴↑𝑗) = (𝐴↑𝑀)) | |
25 | oveq2 7166 | . . . . 5 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁)) | |
26 | geoserg.4 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
27 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
28 | elfzuz 12907 | . . . . . . 7 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ (ℤ≥‘𝑀)) | |
29 | eluznn0 12320 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ ℕ0) | |
30 | 8, 28, 29 | syl2an 597 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℕ0) |
31 | 27, 30 | expcld 13513 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐴↑𝑗) ∈ ℂ) |
32 | 22, 23, 24, 25, 26, 31 | telfsumo 15159 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)((𝐴↑𝑘) − (𝐴↑(𝑘 + 1))) = ((𝐴↑𝑀) − (𝐴↑𝑁))) |
33 | 13, 21, 32 | 3eqtrrd 2863 | . . 3 ⊢ (𝜑 → ((𝐴↑𝑀) − (𝐴↑𝑁)) = (Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) · (1 − 𝐴))) |
34 | 4, 8 | expcld 13513 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℂ) |
35 | eluznn0 12320 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℕ0) | |
36 | 8, 26, 35 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
37 | 4, 36 | expcld 13513 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
38 | 34, 37 | subcld 10999 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑀) − (𝐴↑𝑁)) ∈ ℂ) |
39 | 2, 12 | fsumcl 15092 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) ∈ ℂ) |
40 | geoserg.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 1) | |
41 | 40 | necomd 3073 | . . . . 5 ⊢ (𝜑 → 1 ≠ 𝐴) |
42 | subeq0 10914 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴)) | |
43 | 3, 4, 42 | sylancr 589 | . . . . . 6 ⊢ (𝜑 → ((1 − 𝐴) = 0 ↔ 1 = 𝐴)) |
44 | 43 | necon3bid 3062 | . . . . 5 ⊢ (𝜑 → ((1 − 𝐴) ≠ 0 ↔ 1 ≠ 𝐴)) |
45 | 41, 44 | mpbird 259 | . . . 4 ⊢ (𝜑 → (1 − 𝐴) ≠ 0) |
46 | 38, 39, 6, 45 | divmul3d 11452 | . . 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 259 | . 2 ⊢ (𝜑 → (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴)) = Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘)) |
48 | 47 | eqcomd 2829 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) = (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 − cmin 10872 / cdiv 11299 ℕ0cn0 11900 ℤ≥cuz 12246 ...cfz 12895 ..^cfzo 13036 ↑cexp 13432 Σcsu 15044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 |
This theorem is referenced by: geoser 15224 rplogsumlem2 26063 rpvmasumlem 26065 dchrisum0flblem1 26086 |
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