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Mirrors > Home > MPE Home > Th. List > georeclim | Structured version Visualization version GIF version |
Description: The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Ref | Expression |
---|---|
georeclim.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
georeclim.2 | ⊢ (𝜑 → 1 < (abs‘𝐴)) |
georeclim.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) |
Ref | Expression |
---|---|
georeclim | ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | georeclim.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | georeclim.2 | . . . . 5 ⊢ (𝜑 → 1 < (abs‘𝐴)) | |
3 | 0le1 10551 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
4 | 0re 10040 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
5 | 1re 10039 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | lenlti 10157 | . . . . . . . 8 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
7 | 3, 6 | mpbi 220 | . . . . . . 7 ⊢ ¬ 1 < 0 |
8 | fveq2 6191 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
9 | abs0 14025 | . . . . . . . . 9 ⊢ (abs‘0) = 0 | |
10 | 8, 9 | syl6eq 2672 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
11 | 10 | breq2d 4665 | . . . . . . 7 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
12 | 7, 11 | mtbiri 317 | . . . . . 6 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
13 | 12 | necon2ai 2823 | . . . . 5 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 0) |
15 | 1, 14 | reccld 10794 | . . 3 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
16 | 1cnd 10056 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | 16, 1, 14 | absdivd 14194 | . . . . 5 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴))) |
18 | abs1 14037 | . . . . . 6 ⊢ (abs‘1) = 1 | |
19 | 18 | oveq1i 6660 | . . . . 5 ⊢ ((abs‘1) / (abs‘𝐴)) = (1 / (abs‘𝐴)) |
20 | 17, 19 | syl6eq 2672 | . . . 4 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = (1 / (abs‘𝐴))) |
21 | 1, 14 | absrpcld 14187 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
22 | 21 | recgt1d 11886 | . . . . 5 ⊢ (𝜑 → (1 < (abs‘𝐴) ↔ (1 / (abs‘𝐴)) < 1)) |
23 | 2, 22 | mpbid 222 | . . . 4 ⊢ (𝜑 → (1 / (abs‘𝐴)) < 1) |
24 | 20, 23 | eqbrtrd 4675 | . . 3 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1) |
25 | georeclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) | |
26 | 15, 24, 25 | geolim 14601 | . 2 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − (1 / 𝐴)))) |
27 | 1, 16, 1, 14 | divsubdird 10840 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = ((𝐴 / 𝐴) − (1 / 𝐴))) |
28 | 1, 14 | dividd 10799 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐴) = 1) |
29 | 28 | oveq1d 6665 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 𝐴) − (1 / 𝐴)) = (1 − (1 / 𝐴))) |
30 | 27, 29 | eqtrd 2656 | . . . 4 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = (1 − (1 / 𝐴))) |
31 | 30 | oveq2d 6666 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (1 / (1 − (1 / 𝐴)))) |
32 | ax-1cn 9994 | . . . . 5 ⊢ 1 ∈ ℂ | |
33 | subcl 10280 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ) | |
34 | 1, 32, 33 | sylancl 694 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
35 | 5 | ltnri 10146 | . . . . . . . 8 ⊢ ¬ 1 < 1 |
36 | fveq2 6191 | . . . . . . . . . 10 ⊢ (𝐴 = 1 → (abs‘𝐴) = (abs‘1)) | |
37 | 36, 18 | syl6eq 2672 | . . . . . . . . 9 ⊢ (𝐴 = 1 → (abs‘𝐴) = 1) |
38 | 37 | breq2d 4665 | . . . . . . . 8 ⊢ (𝐴 = 1 → (1 < (abs‘𝐴) ↔ 1 < 1)) |
39 | 35, 38 | mtbiri 317 | . . . . . . 7 ⊢ (𝐴 = 1 → ¬ 1 < (abs‘𝐴)) |
40 | 39 | necon2ai 2823 | . . . . . 6 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 1) |
41 | 2, 40 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1) |
42 | subeq0 10307 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) | |
43 | 1, 32, 42 | sylancl 694 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) |
44 | 43 | necon3bid 2838 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1)) |
45 | 41, 44 | mpbird 247 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ≠ 0) |
46 | 34, 1, 45, 14 | recdivd 10818 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (𝐴 / (𝐴 − 1))) |
47 | 31, 46 | eqtr3d 2658 | . 2 ⊢ (𝜑 → (1 / (1 − (1 / 𝐴))) = (𝐴 / (𝐴 − 1))) |
48 | 26, 47 | breqtrd 4679 | 1 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕ0cn0 11292 seqcseq 12801 ↑cexp 12860 abscabs 13974 ⇝ cli 14215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 |
This theorem is referenced by: geoisumr 14609 ege2le3 14820 eftlub 14839 |
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