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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 11741 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
4 | 0re 11220 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
5 | 1re 11218 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | lenlti 11338 | . . . . . . . 8 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
7 | 3, 6 | mpbi 229 | . . . . . . 7 ⊢ ¬ 1 < 0 |
8 | fveq2 6885 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
9 | abs0 15238 | . . . . . . . . 9 ⊢ (abs‘0) = 0 | |
10 | 8, 9 | eqtrdi 2782 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
11 | 10 | breq2d 5153 | . . . . . . 7 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
12 | 7, 11 | mtbiri 327 | . . . . . 6 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
13 | 12 | necon2ai 2964 | . . . . 5 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 0) |
15 | 1, 14 | reccld 11987 | . . 3 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
16 | 1cnd 11213 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
17 | 16, 1, 14 | absdivd 15408 | . . . . 5 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴))) |
18 | abs1 15250 | . . . . . 6 ⊢ (abs‘1) = 1 | |
19 | 18 | oveq1i 7415 | . . . . 5 ⊢ ((abs‘1) / (abs‘𝐴)) = (1 / (abs‘𝐴)) |
20 | 17, 19 | eqtrdi 2782 | . . . 4 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = (1 / (abs‘𝐴))) |
21 | 1, 14 | absrpcld 15401 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
22 | 21 | recgt1d 13036 | . . . . 5 ⊢ (𝜑 → (1 < (abs‘𝐴) ↔ (1 / (abs‘𝐴)) < 1)) |
23 | 2, 22 | mpbid 231 | . . . 4 ⊢ (𝜑 → (1 / (abs‘𝐴)) < 1) |
24 | 20, 23 | eqbrtrd 5163 | . . 3 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1) |
25 | georeclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) | |
26 | 15, 24, 25 | geolim 15822 | . 2 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − (1 / 𝐴)))) |
27 | 1, 16, 1, 14 | divsubdird 12033 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = ((𝐴 / 𝐴) − (1 / 𝐴))) |
28 | 1, 14 | dividd 11992 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐴) = 1) |
29 | 28 | oveq1d 7420 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 𝐴) − (1 / 𝐴)) = (1 − (1 / 𝐴))) |
30 | 27, 29 | eqtrd 2766 | . . . 4 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = (1 − (1 / 𝐴))) |
31 | 30 | oveq2d 7421 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (1 / (1 − (1 / 𝐴)))) |
32 | ax-1cn 11170 | . . . . 5 ⊢ 1 ∈ ℂ | |
33 | subcl 11463 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ) | |
34 | 1, 32, 33 | sylancl 585 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
35 | 5 | ltnri 11327 | . . . . . . . 8 ⊢ ¬ 1 < 1 |
36 | fveq2 6885 | . . . . . . . . . 10 ⊢ (𝐴 = 1 → (abs‘𝐴) = (abs‘1)) | |
37 | 36, 18 | eqtrdi 2782 | . . . . . . . . 9 ⊢ (𝐴 = 1 → (abs‘𝐴) = 1) |
38 | 37 | breq2d 5153 | . . . . . . . 8 ⊢ (𝐴 = 1 → (1 < (abs‘𝐴) ↔ 1 < 1)) |
39 | 35, 38 | mtbiri 327 | . . . . . . 7 ⊢ (𝐴 = 1 → ¬ 1 < (abs‘𝐴)) |
40 | 39 | necon2ai 2964 | . . . . . 6 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 1) |
41 | 2, 40 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1) |
42 | subeq0 11490 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) | |
43 | 1, 32, 42 | sylancl 585 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) |
44 | 43 | necon3bid 2979 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1)) |
45 | 41, 44 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ≠ 0) |
46 | 34, 1, 45, 14 | recdivd 12011 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (𝐴 / (𝐴 − 1))) |
47 | 31, 46 | eqtr3d 2768 | . 2 ⊢ (𝜑 → (1 / (1 − (1 / 𝐴))) = (𝐴 / (𝐴 − 1))) |
48 | 26, 47 | breqtrd 5167 | 1 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 ℕ0cn0 12476 seqcseq 13972 ↑cexp 14032 abscabs 15187 ⇝ cli 15434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639 |
This theorem is referenced by: geoisumr 15830 ege2le3 16040 eftlub 16059 |
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