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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 10430 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
4 | 0re 9919 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
5 | 1re 9918 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | lenlti 10036 | . . . . . . . 8 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
7 | 3, 6 | mpbi 219 | . . . . . . 7 ⊢ ¬ 1 < 0 |
8 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
9 | abs0 13873 | . . . . . . . . 9 ⊢ (abs‘0) = 0 | |
10 | 8, 9 | syl6eq 2660 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
11 | 10 | breq2d 4595 | . . . . . . 7 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
12 | 7, 11 | mtbiri 316 | . . . . . 6 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
13 | 12 | necon2ai 2811 | . . . . 5 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 0) |
15 | 1, 14 | reccld 10673 | . . 3 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
16 | 1cnd 9935 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | 16, 1, 14 | absdivd 14042 | . . . . 5 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴))) |
18 | abs1 13885 | . . . . . 6 ⊢ (abs‘1) = 1 | |
19 | 18 | oveq1i 6559 | . . . . 5 ⊢ ((abs‘1) / (abs‘𝐴)) = (1 / (abs‘𝐴)) |
20 | 17, 19 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = (1 / (abs‘𝐴))) |
21 | 1, 14 | absrpcld 14035 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
22 | 21 | recgt1d 11762 | . . . . 5 ⊢ (𝜑 → (1 < (abs‘𝐴) ↔ (1 / (abs‘𝐴)) < 1)) |
23 | 2, 22 | mpbid 221 | . . . 4 ⊢ (𝜑 → (1 / (abs‘𝐴)) < 1) |
24 | 20, 23 | eqbrtrd 4605 | . . 3 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1) |
25 | georeclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) | |
26 | 15, 24, 25 | geolim 14440 | . 2 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − (1 / 𝐴)))) |
27 | 1, 16, 1, 14 | divsubdird 10719 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = ((𝐴 / 𝐴) − (1 / 𝐴))) |
28 | 1, 14 | dividd 10678 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐴) = 1) |
29 | 28 | oveq1d 6564 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 𝐴) − (1 / 𝐴)) = (1 − (1 / 𝐴))) |
30 | 27, 29 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = (1 − (1 / 𝐴))) |
31 | 30 | oveq2d 6565 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (1 / (1 − (1 / 𝐴)))) |
32 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
33 | subcl 10159 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ) | |
34 | 1, 32, 33 | sylancl 693 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
35 | 5 | ltnri 10025 | . . . . . . . 8 ⊢ ¬ 1 < 1 |
36 | fveq2 6103 | . . . . . . . . . 10 ⊢ (𝐴 = 1 → (abs‘𝐴) = (abs‘1)) | |
37 | 36, 18 | syl6eq 2660 | . . . . . . . . 9 ⊢ (𝐴 = 1 → (abs‘𝐴) = 1) |
38 | 37 | breq2d 4595 | . . . . . . . 8 ⊢ (𝐴 = 1 → (1 < (abs‘𝐴) ↔ 1 < 1)) |
39 | 35, 38 | mtbiri 316 | . . . . . . 7 ⊢ (𝐴 = 1 → ¬ 1 < (abs‘𝐴)) |
40 | 39 | necon2ai 2811 | . . . . . 6 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 1) |
41 | 2, 40 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1) |
42 | subeq0 10186 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) | |
43 | 1, 32, 42 | sylancl 693 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) |
44 | 43 | necon3bid 2826 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1)) |
45 | 41, 44 | mpbird 246 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ≠ 0) |
46 | 34, 1, 45, 14 | recdivd 10697 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (𝐴 / (𝐴 − 1))) |
47 | 31, 46 | eqtr3d 2646 | . 2 ⊢ (𝜑 → (1 / (1 − (1 / 𝐴))) = (𝐴 / (𝐴 − 1))) |
48 | 26, 47 | breqtrd 4609 | 1 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕ0cn0 11169 seqcseq 12663 ↑cexp 12722 abscabs 13822 ⇝ cli 14063 |
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: geoisumr 14448 ege2le3 14659 eftlub 14678 |
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