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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climdivf | Structured version Visualization version GIF version |
Description: Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
climdivf.1 | ⊢ Ⅎ𝑘𝜑 |
climdivf.2 | ⊢ Ⅎ𝑘𝐹 |
climdivf.3 | ⊢ Ⅎ𝑘𝐺 |
climdivf.4 | ⊢ Ⅎ𝑘𝐻 |
climdivf.5 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climdivf.6 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climdivf.7 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climdivf.8 | ⊢ (𝜑 → 𝐻 ∈ 𝑋) |
climdivf.9 | ⊢ (𝜑 → 𝐺 ⇝ 𝐵) |
climdivf.10 | ⊢ (𝜑 → 𝐵 ≠ 0) |
climdivf.11 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
climdivf.12 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) |
climdivf.13 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘))) |
Ref | Expression |
---|---|
climdivf | ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climdivf.1 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | climdivf.2 | . . 3 ⊢ Ⅎ𝑘𝐹 | |
3 | nfmpt1 5128 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) | |
4 | climdivf.4 | . . 3 ⊢ Ⅎ𝑘𝐻 | |
5 | climdivf.5 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | climdivf.6 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | climdivf.7 | . . 3 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
8 | climdivf.8 | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑋) | |
9 | climdivf.3 | . . . 4 ⊢ Ⅎ𝑘𝐺 | |
10 | climdivf.9 | . . . 4 ⊢ (𝜑 → 𝐺 ⇝ 𝐵) | |
11 | climdivf.10 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 0) | |
12 | climdivf.12 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) | |
13 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
14 | 12 | eldifad 3893 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
15 | eldifsni 4683 | . . . . . . 7 ⊢ ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) → (𝐺‘𝑘) ≠ 0) | |
16 | 12, 15 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≠ 0) |
17 | 14, 16 | reccld 11398 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) ∈ ℂ) |
18 | eqid 2798 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) | |
19 | 18 | fvmpt2 6756 | . . . . 5 ⊢ ((𝑘 ∈ 𝑍 ∧ (1 / (𝐺‘𝑘)) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘) = (1 / (𝐺‘𝑘))) |
20 | 13, 17, 19 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘) = (1 / (𝐺‘𝑘))) |
21 | 5 | fvexi 6659 | . . . . . 6 ⊢ 𝑍 ∈ V |
22 | 21 | mptex 6963 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) ∈ V |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) ∈ V) |
24 | 1, 9, 3, 5, 6, 10, 11, 12, 20, 23 | climrecf 42251 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) ⇝ (1 / 𝐵)) |
25 | climdivf.11 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
26 | 20, 17 | eqeltrd 2890 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘) ∈ ℂ) |
27 | climdivf.13 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘))) | |
28 | 25, 14, 16 | divrecd 11408 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) / (𝐺‘𝑘)) = ((𝐹‘𝑘) · (1 / (𝐺‘𝑘)))) |
29 | 20 | eqcomd 2804 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) = ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘)) |
30 | 29 | oveq2d 7151 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) · (1 / (𝐺‘𝑘))) = ((𝐹‘𝑘) · ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘))) |
31 | 27, 28, 30 | 3eqtrd 2837 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘))) |
32 | 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 31 | climmulf 42246 | . 2 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · (1 / 𝐵))) |
33 | climcl 14848 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
34 | 7, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
35 | climcl 14848 | . . . 4 ⊢ (𝐺 ⇝ 𝐵 → 𝐵 ∈ ℂ) | |
36 | 10, 35 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
37 | 34, 36, 11 | divrecd 11408 | . 2 ⊢ (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
38 | 32, 37 | breqtrrd 5058 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 · cmul 10531 / cdiv 11286 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 |
This theorem is referenced by: stirlinglem8 42723 fourierdlem103 42851 fourierdlem104 42852 |
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