Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > geoisum1c | Structured version Visualization version GIF version |
Description: The infinite sum of 𝐴 · (𝑅↑1) + 𝐴 · (𝑅↑2)... is (𝐴 · 𝑅) / (1 − 𝑅). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Ref | Expression |
---|---|
geoisum1c | ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 𝐴 ∈ ℂ) | |
2 | simp2 1055 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 𝑅 ∈ ℂ) | |
3 | ax-1cn 9873 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | subcl 10159 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (1 − 𝑅) ∈ ℂ) | |
5 | 3, 2, 4 | sylancr 694 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (1 − 𝑅) ∈ ℂ) |
6 | simp3 1056 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (abs‘𝑅) < 1) | |
7 | 1re 9918 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
8 | 7 | ltnri 10025 | . . . . . . 7 ⊢ ¬ 1 < 1 |
9 | abs1 13885 | . . . . . . . . 9 ⊢ (abs‘1) = 1 | |
10 | fveq2 6103 | . . . . . . . . 9 ⊢ (1 = 𝑅 → (abs‘1) = (abs‘𝑅)) | |
11 | 9, 10 | syl5eqr 2658 | . . . . . . . 8 ⊢ (1 = 𝑅 → 1 = (abs‘𝑅)) |
12 | 11 | breq1d 4593 | . . . . . . 7 ⊢ (1 = 𝑅 → (1 < 1 ↔ (abs‘𝑅) < 1)) |
13 | 8, 12 | mtbii 315 | . . . . . 6 ⊢ (1 = 𝑅 → ¬ (abs‘𝑅) < 1) |
14 | 13 | necon2ai 2811 | . . . . 5 ⊢ ((abs‘𝑅) < 1 → 1 ≠ 𝑅) |
15 | 6, 14 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ≠ 𝑅) |
16 | subeq0 10186 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑅 ∈ ℂ) → ((1 − 𝑅) = 0 ↔ 1 = 𝑅)) | |
17 | 16 | necon3bid 2826 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝑅 ∈ ℂ) → ((1 − 𝑅) ≠ 0 ↔ 1 ≠ 𝑅)) |
18 | 3, 2, 17 | sylancr 694 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → ((1 − 𝑅) ≠ 0 ↔ 1 ≠ 𝑅)) |
19 | 15, 18 | mpbird 246 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (1 − 𝑅) ≠ 0) |
20 | 1, 2, 5, 19 | divassd 10715 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → ((𝐴 · 𝑅) / (1 − 𝑅)) = (𝐴 · (𝑅 / (1 − 𝑅)))) |
21 | geoisum1 14449 | . . . 4 ⊢ ((𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝑅↑𝑘) = (𝑅 / (1 − 𝑅))) | |
22 | 21 | 3adant1 1072 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝑅↑𝑘) = (𝑅 / (1 − 𝑅))) |
23 | 22 | oveq2d 6565 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (𝐴 · Σ𝑘 ∈ ℕ (𝑅↑𝑘)) = (𝐴 · (𝑅 / (1 − 𝑅)))) |
24 | nnuz 11599 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
25 | 1zzd 11285 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℤ) | |
26 | oveq2 6557 | . . . . 5 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑛) = (𝑅↑𝑘)) | |
27 | eqid 2610 | . . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (𝑅↑𝑛)) = (𝑛 ∈ ℕ ↦ (𝑅↑𝑛)) | |
28 | ovex 6577 | . . . . 5 ⊢ (𝑅↑𝑘) ∈ V | |
29 | 26, 27, 28 | fvmpt 6191 | . . . 4 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑅↑𝑛))‘𝑘) = (𝑅↑𝑘)) |
30 | 29 | adantl 481 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑅↑𝑛))‘𝑘) = (𝑅↑𝑘)) |
31 | nnnn0 11176 | . . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
32 | expcl 12740 | . . . 4 ⊢ ((𝑅 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑅↑𝑘) ∈ ℂ) | |
33 | 2, 31, 32 | syl2an 493 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑘) ∈ ℂ) |
34 | 1nn0 11185 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
35 | 34 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℕ0) |
36 | elnnuz 11600 | . . . . . 6 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1)) | |
37 | 36, 30 | sylan2br 492 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝑛 ∈ ℕ ↦ (𝑅↑𝑛))‘𝑘) = (𝑅↑𝑘)) |
38 | 2, 6, 35, 37 | geolim2 14441 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ⇝ ((𝑅↑1) / (1 − 𝑅))) |
39 | seqex 12665 | . . . . 5 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ V | |
40 | ovex 6577 | . . . . 5 ⊢ ((𝑅↑1) / (1 − 𝑅)) ∈ V | |
41 | 39, 40 | breldm 5251 | . . . 4 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ⇝ ((𝑅↑1) / (1 − 𝑅)) → seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ dom ⇝ ) |
42 | 38, 41 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ dom ⇝ ) |
43 | 24, 25, 30, 33, 42, 1 | isummulc2 14335 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (𝐴 · Σ𝑘 ∈ ℕ (𝑅↑𝑘)) = Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘))) |
44 | 20, 23, 43 | 3eqtr2rd 2651 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 − cmin 10145 / cdiv 10563 ℕcn 10897 ℕ0cn0 11169 ℤ≥cuz 11563 seqcseq 12663 ↑cexp 12722 abscabs 13822 ⇝ cli 14063 Σcsu 14264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 |
This theorem is referenced by: 0.999... 14451 0.999...OLD 14452 |
Copyright terms: Public domain | W3C validator |