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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 11157 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
4 | 0re 10637 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
5 | 1re 10635 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | lenlti 10754 | . . . . . . . 8 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
7 | 3, 6 | mpbi 232 | . . . . . . 7 ⊢ ¬ 1 < 0 |
8 | fveq2 6665 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
9 | abs0 14639 | . . . . . . . . 9 ⊢ (abs‘0) = 0 | |
10 | 8, 9 | syl6eq 2872 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
11 | 10 | breq2d 5071 | . . . . . . 7 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
12 | 7, 11 | mtbiri 329 | . . . . . 6 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
13 | 12 | necon2ai 3045 | . . . . 5 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 0) |
15 | 1, 14 | reccld 11403 | . . 3 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
16 | 1cnd 10630 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | 16, 1, 14 | absdivd 14809 | . . . . 5 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴))) |
18 | abs1 14651 | . . . . . 6 ⊢ (abs‘1) = 1 | |
19 | 18 | oveq1i 7160 | . . . . 5 ⊢ ((abs‘1) / (abs‘𝐴)) = (1 / (abs‘𝐴)) |
20 | 17, 19 | syl6eq 2872 | . . . 4 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = (1 / (abs‘𝐴))) |
21 | 1, 14 | absrpcld 14802 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
22 | 21 | recgt1d 12439 | . . . . 5 ⊢ (𝜑 → (1 < (abs‘𝐴) ↔ (1 / (abs‘𝐴)) < 1)) |
23 | 2, 22 | mpbid 234 | . . . 4 ⊢ (𝜑 → (1 / (abs‘𝐴)) < 1) |
24 | 20, 23 | eqbrtrd 5081 | . . 3 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1) |
25 | georeclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) | |
26 | 15, 24, 25 | geolim 15220 | . 2 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − (1 / 𝐴)))) |
27 | 1, 16, 1, 14 | divsubdird 11449 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = ((𝐴 / 𝐴) − (1 / 𝐴))) |
28 | 1, 14 | dividd 11408 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐴) = 1) |
29 | 28 | oveq1d 7165 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 𝐴) − (1 / 𝐴)) = (1 − (1 / 𝐴))) |
30 | 27, 29 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = (1 − (1 / 𝐴))) |
31 | 30 | oveq2d 7166 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (1 / (1 − (1 / 𝐴)))) |
32 | ax-1cn 10589 | . . . . 5 ⊢ 1 ∈ ℂ | |
33 | subcl 10879 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ) | |
34 | 1, 32, 33 | sylancl 588 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
35 | 5 | ltnri 10743 | . . . . . . . 8 ⊢ ¬ 1 < 1 |
36 | fveq2 6665 | . . . . . . . . . 10 ⊢ (𝐴 = 1 → (abs‘𝐴) = (abs‘1)) | |
37 | 36, 18 | syl6eq 2872 | . . . . . . . . 9 ⊢ (𝐴 = 1 → (abs‘𝐴) = 1) |
38 | 37 | breq2d 5071 | . . . . . . . 8 ⊢ (𝐴 = 1 → (1 < (abs‘𝐴) ↔ 1 < 1)) |
39 | 35, 38 | mtbiri 329 | . . . . . . 7 ⊢ (𝐴 = 1 → ¬ 1 < (abs‘𝐴)) |
40 | 39 | necon2ai 3045 | . . . . . 6 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 1) |
41 | 2, 40 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1) |
42 | subeq0 10906 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) | |
43 | 1, 32, 42 | sylancl 588 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) |
44 | 43 | necon3bid 3060 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1)) |
45 | 41, 44 | mpbird 259 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ≠ 0) |
46 | 34, 1, 45, 14 | recdivd 11427 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (𝐴 / (𝐴 − 1))) |
47 | 31, 46 | eqtr3d 2858 | . 2 ⊢ (𝜑 → (1 / (1 − (1 / 𝐴))) = (𝐴 / (𝐴 − 1))) |
48 | 26, 47 | breqtrd 5085 | 1 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 − cmin 10864 / cdiv 11291 ℕ0cn0 11891 seqcseq 13363 ↑cexp 13423 abscabs 14587 ⇝ cli 14835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 |
This theorem is referenced by: geoisumr 15228 ege2le3 15437 eftlub 15456 |
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