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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 11725 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
| 4 | 0re 11198 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
| 5 | 1re 11196 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
| 6 | 4, 5 | lenlti 11318 | . . . . . . . 8 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
| 7 | 3, 6 | mpbi 233 | . . . . . . 7 ⊢ ¬ 1 < 0 |
| 8 | fveq2 6871 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
| 9 | abs0 15324 | . . . . . . . . 9 ⊢ (abs‘0) = 0 | |
| 10 | 8, 9 | eqtrdi 2816 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
| 11 | 10 | breq2d 5116 | . . . . . . 7 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
| 12 | 7, 11 | mtbiri 330 | . . . . . 6 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
| 13 | 12 | necon2ai 2989 | . . . . 5 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
| 14 | 2, 13 | syl 18 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 0) |
| 15 | 1, 14 | reccld 11972 | . . 3 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
| 16 | 1cnd 11190 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
| 17 | 16, 1, 14 | absdivd 15497 | . . . . 5 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴))) |
| 18 | abs1 15336 | . . . . . 6 ⊢ (abs‘1) = 1 | |
| 19 | 18 | oveq1i 7410 | . . . . 5 ⊢ ((abs‘1) / (abs‘𝐴)) = (1 / (abs‘𝐴)) |
| 20 | 17, 19 | eqtrdi 2816 | . . . 4 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = (1 / (abs‘𝐴))) |
| 21 | 1, 14 | absrpcld 15490 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
| 22 | 21 | recgt1d 13062 | . . . . 5 ⊢ (𝜑 → (1 < (abs‘𝐴) ↔ (1 / (abs‘𝐴)) < 1)) |
| 23 | 2, 22 | mpbid 235 | . . . 4 ⊢ (𝜑 → (1 / (abs‘𝐴)) < 1) |
| 24 | 20, 23 | eqbrtrd 5126 | . . 3 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1) |
| 25 | georeclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) | |
| 26 | 15, 24, 25 | geolim 15912 | . 2 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − (1 / 𝐴)))) |
| 27 | 1, 16, 1, 14 | divsubdird 12018 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = ((𝐴 / 𝐴) − (1 / 𝐴))) |
| 28 | 1, 14 | dividd 11977 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐴) = 1) |
| 29 | 28 | oveq1d 7415 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 𝐴) − (1 / 𝐴)) = (1 − (1 / 𝐴))) |
| 30 | 27, 29 | eqtrd 2800 | . . . 4 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = (1 − (1 / 𝐴))) |
| 31 | 30 | oveq2d 7416 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (1 / (1 − (1 / 𝐴)))) |
| 32 | ax-1cn 11146 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 33 | subcl 11444 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ) | |
| 34 | 1, 32, 33 | sylancl 597 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
| 35 | 5 | ltnri 11307 | . . . . . . . 8 ⊢ ¬ 1 < 1 |
| 36 | fveq2 6871 | . . . . . . . . . 10 ⊢ (𝐴 = 1 → (abs‘𝐴) = (abs‘1)) | |
| 37 | 36, 18 | eqtrdi 2816 | . . . . . . . . 9 ⊢ (𝐴 = 1 → (abs‘𝐴) = 1) |
| 38 | 37 | breq2d 5116 | . . . . . . . 8 ⊢ (𝐴 = 1 → (1 < (abs‘𝐴) ↔ 1 < 1)) |
| 39 | 35, 38 | mtbiri 330 | . . . . . . 7 ⊢ (𝐴 = 1 → ¬ 1 < (abs‘𝐴)) |
| 40 | 39 | necon2ai 2989 | . . . . . 6 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 1) |
| 41 | 2, 40 | syl 18 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1) |
| 42 | subeq0 11472 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) | |
| 43 | 1, 32, 42 | sylancl 597 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) |
| 44 | 43 | necon3bid 3004 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1)) |
| 45 | 41, 44 | mpbird 260 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ≠ 0) |
| 46 | 34, 1, 45, 14 | recdivd 11996 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (𝐴 / (𝐴 − 1))) |
| 47 | 31, 46 | eqtr3d 2802 | . 2 ⊢ (𝜑 → (1 / (1 − (1 / 𝐴))) = (𝐴 / (𝐴 − 1))) |
| 48 | 26, 47 | breqtrd 5130 | 1 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 class class class wbr 5104 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 0cc0 11088 1c1 11089 + caddc 11091 < clt 11231 ≤ cle 11232 − cmin 11429 / cdiv 11859 ℕ0cn0 12492 seqcseq 14025 ↑cexp 14085 abscabs 15273 ⇝ cli 15523 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4908 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-pm 8815 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-n0 12493 df-z 12580 df-uz 12851 df-rp 13005 df-fz 13524 df-fzo 13671 df-fl 13813 df-seq 14026 df-exp 14086 df-hash 14355 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15527 df-rlim 15528 df-sum 15726 |
| This theorem is referenced by: geoisumr 15920 ege2le3 16132 eftlub 16153 |
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