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 5155 | . . 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 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
14 | 12 | eldifad 3945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
15 | eldifsni 4714 | . . . . . . 7 ⊢ ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) → (𝐺‘𝑘) ≠ 0) | |
16 | 12, 15 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≠ 0) |
17 | 14, 16 | reccld 11397 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) ∈ ℂ) |
18 | eqid 2818 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) | |
19 | 18 | fvmpt2 6771 | . . . . 5 ⊢ ((𝑘 ∈ 𝑍 ∧ (1 / (𝐺‘𝑘)) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘) = (1 / (𝐺‘𝑘))) |
20 | 13, 17, 19 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘) = (1 / (𝐺‘𝑘))) |
21 | 5 | fvexi 6677 | . . . . . 6 ⊢ 𝑍 ∈ V |
22 | 21 | mptex 6977 | . . . . 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 41766 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) ⇝ (1 / 𝐵)) |
25 | climdivf.11 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
26 | 20, 17 | eqeltrd 2910 | . . 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 11407 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) / (𝐺‘𝑘)) = ((𝐹‘𝑘) · (1 / (𝐺‘𝑘)))) |
29 | 20 | eqcomd 2824 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) = ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘)) |
30 | 29 | oveq2d 7161 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) · (1 / (𝐺‘𝑘))) = ((𝐹‘𝑘) · ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘))) |
31 | 27, 28, 30 | 3eqtrd 2857 | . . 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 41761 | . 2 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · (1 / 𝐵))) |
33 | climcl 14844 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
34 | 7, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
35 | climcl 14844 | . . . 4 ⊢ (𝐺 ⇝ 𝐵 → 𝐵 ∈ ℂ) | |
36 | 10, 35 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
37 | 34, 36, 11 | divrecd 11407 | . 2 ⊢ (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
38 | 32, 37 | breqtrrd 5085 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 ≠ wne 3013 Vcvv 3492 ∖ cdif 3930 {csn 4557 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 · cmul 10530 / cdiv 11285 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 |
This theorem is referenced by: stirlinglem8 42243 fourierdlem103 42371 fourierdlem104 42372 |
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