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Mirrors > Home > ILE Home > Th. List > georeclim | 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 | 1 | abscld 10745 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ) |
3 | 0red 7586 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
4 | 1red 7600 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
5 | 0lt1 7707 | . . . . . . . 8 ⊢ 0 < 1 | |
6 | 5 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 0 < 1) |
7 | georeclim.2 | . . . . . . 7 ⊢ (𝜑 → 1 < (abs‘𝐴)) | |
8 | 3, 4, 2, 6, 7 | lttrd 7706 | . . . . . 6 ⊢ (𝜑 → 0 < (abs‘𝐴)) |
9 | 2, 8 | gt0ap0d 8202 | . . . . 5 ⊢ (𝜑 → (abs‘𝐴) # 0) |
10 | abs00ap 10626 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) # 0 ↔ 𝐴 # 0)) | |
11 | 1, 10 | syl 14 | . . . . 5 ⊢ (𝜑 → ((abs‘𝐴) # 0 ↔ 𝐴 # 0)) |
12 | 9, 11 | mpbid 146 | . . . 4 ⊢ (𝜑 → 𝐴 # 0) |
13 | 1, 12 | recclapd 8345 | . . 3 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
14 | 1cnd 7601 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
15 | 14, 1, 12 | absdivapd 10759 | . . . . 5 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴))) |
16 | abs1 10636 | . . . . . 6 ⊢ (abs‘1) = 1 | |
17 | 16 | oveq1i 5700 | . . . . 5 ⊢ ((abs‘1) / (abs‘𝐴)) = (1 / (abs‘𝐴)) |
18 | 15, 17 | syl6eq 2143 | . . . 4 ⊢ (𝜑 → (abs‘(1 / 𝐴)) = (1 / (abs‘𝐴))) |
19 | 2, 8 | elrpd 9270 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
20 | 19 | recgt1d 9287 | . . . . 5 ⊢ (𝜑 → (1 < (abs‘𝐴) ↔ (1 / (abs‘𝐴)) < 1)) |
21 | 7, 20 | mpbid 146 | . . . 4 ⊢ (𝜑 → (1 / (abs‘𝐴)) < 1) |
22 | 18, 21 | eqbrtrd 3887 | . . 3 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1) |
23 | georeclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) | |
24 | 13, 22, 23 | geolim 11070 | . 2 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − (1 / 𝐴)))) |
25 | 1, 14, 1, 12 | divsubdirapd 8394 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = ((𝐴 / 𝐴) − (1 / 𝐴))) |
26 | 1, 12 | dividapd 8350 | . . . . . 6 ⊢ (𝜑 → (𝐴 / 𝐴) = 1) |
27 | 26 | oveq1d 5705 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 𝐴) − (1 / 𝐴)) = (1 − (1 / 𝐴))) |
28 | 25, 27 | eqtrd 2127 | . . . 4 ⊢ (𝜑 → ((𝐴 − 1) / 𝐴) = (1 − (1 / 𝐴))) |
29 | 28 | oveq2d 5706 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (1 / (1 − (1 / 𝐴)))) |
30 | ax-1cn 7535 | . . . . 5 ⊢ 1 ∈ ℂ | |
31 | subcl 7778 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ) | |
32 | 1, 30, 31 | sylancl 405 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ) |
33 | 4, 6 | elrpd 9270 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ+) |
34 | 1, 33, 7 | absgtap 11069 | . . . . 5 ⊢ (𝜑 → 𝐴 # 1) |
35 | 1, 14, 34 | subap0d 8216 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) # 0) |
36 | 32, 1, 35, 12 | recdivapd 8371 | . . 3 ⊢ (𝜑 → (1 / ((𝐴 − 1) / 𝐴)) = (𝐴 / (𝐴 − 1))) |
37 | 29, 36 | eqtr3d 2129 | . 2 ⊢ (𝜑 → (1 / (1 − (1 / 𝐴))) = (𝐴 / (𝐴 − 1))) |
38 | 24, 37 | breqtrd 3891 | 1 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 ℂcc 7445 0cc0 7447 1c1 7448 + caddc 7450 < clt 7619 − cmin 7750 # cap 8155 / cdiv 8236 ℕ0cn0 8771 seqcseq 10001 ↑cexp 10085 abscabs 10561 ⇝ cli 10837 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 ax-caucvg 7562 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-isom 5058 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-irdg 6173 df-frec 6194 df-1o 6219 df-oadd 6223 df-er 6332 df-en 6538 df-dom 6539 df-fin 6540 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-n0 8772 df-z 8849 df-uz 9119 df-q 9204 df-rp 9234 df-fz 9574 df-fzo 9703 df-iseq 10002 df-seq3 10003 df-exp 10086 df-ihash 10315 df-cj 10407 df-re 10408 df-im 10409 df-rsqrt 10562 df-abs 10563 df-clim 10838 df-sumdc 10913 |
This theorem is referenced by: geoisumr 11077 ege2le3 11126 eftlub 11145 |
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