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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 11786 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
| 4 | 0re 11263 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
| 5 | 1re 11261 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
| 6 | 4, 5 | lenlti 11381 | . . . . . . . 8 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) | 
| 7 | 3, 6 | mpbi 230 | . . . . . . 7 ⊢ ¬ 1 < 0 | 
| 8 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
| 9 | abs0 15324 | . . . . . . . . 9 ⊢ (abs‘0) = 0 | |
| 10 | 8, 9 | eqtrdi 2793 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) | 
| 11 | 10 | breq2d 5155 | . . . . . . 7 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) | 
| 12 | 7, 11 | mtbiri 327 | . . . . . 6 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) | 
| 13 | 12 | necon2ai 2970 | . . . . 5 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) | 
| 14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 0) | 
| 15 | 1, 14 | reccld 12036 | . . 3 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) | 
| 16 | 1cnd 11256 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
| 17 | 16, 1, 14 | absdivd 15494 | . . . . 5 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴))) | 
| 18 | abs1 15336 | . . . . . 6 ⊢ (abs‘1) = 1 | |
| 19 | 18 | oveq1i 7441 | . . . . 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 15487 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) | 
| 22 | 21 | recgt1d 13091 | . . . . 5 ⊢ (𝜑 → (1 < (abs‘𝐴) ↔ (1 / (abs‘𝐴)) < 1)) | 
| 23 | 2, 22 | mpbid 232 | . . . 4 ⊢ (𝜑 → (1 / (abs‘𝐴)) < 1) | 
| 24 | 20, 23 | eqbrtrd 5165 | . . 3 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1) | 
| 25 | georeclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) | |
| 26 | 15, 24, 25 | geolim 15906 | . 2 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − (1 / 𝐴)))) | 
| 27 | 1, 16, 1, 14 | divsubdird 12082 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = ((𝐴 / 𝐴) − (1 / 𝐴))) | 
| 28 | 1, 14 | dividd 12041 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐴) = 1) | 
| 29 | 28 | oveq1d 7446 | . . . . 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 7447 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (1 / (1 − (1 / 𝐴)))) | 
| 32 | ax-1cn 11213 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 33 | subcl 11507 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ) | |
| 34 | 1, 32, 33 | sylancl 586 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) | 
| 35 | 5 | ltnri 11370 | . . . . . . . 8 ⊢ ¬ 1 < 1 | 
| 36 | fveq2 6906 | . . . . . . . . . 10 ⊢ (𝐴 = 1 → (abs‘𝐴) = (abs‘1)) | |
| 37 | 36, 18 | eqtrdi 2793 | . . . . . . . . 9 ⊢ (𝐴 = 1 → (abs‘𝐴) = 1) | 
| 38 | 37 | breq2d 5155 | . . . . . . . 8 ⊢ (𝐴 = 1 → (1 < (abs‘𝐴) ↔ 1 < 1)) | 
| 39 | 35, 38 | mtbiri 327 | . . . . . . 7 ⊢ (𝐴 = 1 → ¬ 1 < (abs‘𝐴)) | 
| 40 | 39 | necon2ai 2970 | . . . . . 6 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 1) | 
| 41 | 2, 40 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1) | 
| 42 | subeq0 11535 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) | |
| 43 | 1, 32, 42 | sylancl 586 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) = 0 ↔ 𝐴 = 1)) | 
| 44 | 43 | necon3bid 2985 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1)) | 
| 45 | 41, 44 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ≠ 0) | 
| 46 | 34, 1, 45, 14 | recdivd 12060 | . . 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 5169 | 1 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕ0cn0 12526 seqcseq 14042 ↑cexp 14102 abscabs 15273 ⇝ cli 15520 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 | 
| This theorem is referenced by: geoisumr 15914 ege2le3 16126 eftlub 16145 | 
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