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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 11685 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
4 | 0re 11164 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
5 | 1re 11162 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | lenlti 11282 | . . . . . . . 8 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
7 | 3, 6 | mpbi 229 | . . . . . . 7 ⊢ ¬ 1 < 0 |
8 | fveq2 6847 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
9 | abs0 15177 | . . . . . . . . 9 ⊢ (abs‘0) = 0 | |
10 | 8, 9 | eqtrdi 2793 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
11 | 10 | breq2d 5122 | . . . . . . 7 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
12 | 7, 11 | mtbiri 327 | . . . . . 6 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
13 | 12 | necon2ai 2974 | . . . . 5 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 0) |
15 | 1, 14 | reccld 11931 | . . 3 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
16 | 1cnd 11157 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | 16, 1, 14 | absdivd 15347 | . . . . 5 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴))) |
18 | abs1 15189 | . . . . . 6 ⊢ (abs‘1) = 1 | |
19 | 18 | oveq1i 7372 | . . . . 5 ⊢ ((abs‘1) / (abs‘𝐴)) = (1 / (abs‘𝐴)) |
20 | 17, 19 | eqtrdi 2793 | . . . 4 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = (1 / (abs‘𝐴))) |
21 | 1, 14 | absrpcld 15340 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
22 | 21 | recgt1d 12978 | . . . . 5 ⊢ (𝜑 → (1 < (abs‘𝐴) ↔ (1 / (abs‘𝐴)) < 1)) |
23 | 2, 22 | mpbid 231 | . . . 4 ⊢ (𝜑 → (1 / (abs‘𝐴)) < 1) |
24 | 20, 23 | eqbrtrd 5132 | . . 3 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1) |
25 | georeclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) | |
26 | 15, 24, 25 | geolim 15762 | . 2 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − (1 / 𝐴)))) |
27 | 1, 16, 1, 14 | divsubdird 11977 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = ((𝐴 / 𝐴) − (1 / 𝐴))) |
28 | 1, 14 | dividd 11936 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐴) = 1) |
29 | 28 | oveq1d 7377 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 𝐴) − (1 / 𝐴)) = (1 − (1 / 𝐴))) |
30 | 27, 29 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = (1 − (1 / 𝐴))) |
31 | 30 | oveq2d 7378 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (1 / (1 − (1 / 𝐴)))) |
32 | ax-1cn 11116 | . . . . 5 ⊢ 1 ∈ ℂ | |
33 | subcl 11407 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ) | |
34 | 1, 32, 33 | sylancl 587 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
35 | 5 | ltnri 11271 | . . . . . . . 8 ⊢ ¬ 1 < 1 |
36 | fveq2 6847 | . . . . . . . . . 10 ⊢ (𝐴 = 1 → (abs‘𝐴) = (abs‘1)) | |
37 | 36, 18 | eqtrdi 2793 | . . . . . . . . 9 ⊢ (𝐴 = 1 → (abs‘𝐴) = 1) |
38 | 37 | breq2d 5122 | . . . . . . . 8 ⊢ (𝐴 = 1 → (1 < (abs‘𝐴) ↔ 1 < 1)) |
39 | 35, 38 | mtbiri 327 | . . . . . . 7 ⊢ (𝐴 = 1 → ¬ 1 < (abs‘𝐴)) |
40 | 39 | necon2ai 2974 | . . . . . 6 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 1) |
41 | 2, 40 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1) |
42 | subeq0 11434 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) | |
43 | 1, 32, 42 | sylancl 587 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) |
44 | 43 | necon3bid 2989 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1)) |
45 | 41, 44 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ≠ 0) |
46 | 34, 1, 45, 14 | recdivd 11955 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (𝐴 / (𝐴 − 1))) |
47 | 31, 46 | eqtr3d 2779 | . 2 ⊢ (𝜑 → (1 / (1 − (1 / 𝐴))) = (𝐴 / (𝐴 − 1))) |
48 | 26, 47 | breqtrd 5136 | 1 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ℂcc 11056 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 ≤ cle 11197 − cmin 11392 / cdiv 11819 ℕ0cn0 12420 seqcseq 13913 ↑cexp 13974 abscabs 15126 ⇝ cli 15373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 |
This theorem is referenced by: geoisumr 15770 ege2le3 15979 eftlub 15998 |
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