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 5166 | . . 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 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
14 | 12 | eldifad 3950 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
15 | eldifsni 4724 | . . . . . . 7 ⊢ ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) → (𝐺‘𝑘) ≠ 0) | |
16 | 12, 15 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≠ 0) |
17 | 14, 16 | reccld 11411 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) ∈ ℂ) |
18 | eqid 2823 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) | |
19 | 18 | fvmpt2 6781 | . . . . 5 ⊢ ((𝑘 ∈ 𝑍 ∧ (1 / (𝐺‘𝑘)) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘) = (1 / (𝐺‘𝑘))) |
20 | 13, 17, 19 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘) = (1 / (𝐺‘𝑘))) |
21 | 5 | fvexi 6686 | . . . . . 6 ⊢ 𝑍 ∈ V |
22 | 21 | mptex 6988 | . . . . 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 41897 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) ⇝ (1 / 𝐵)) |
25 | climdivf.11 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
26 | 20, 17 | eqeltrd 2915 | . . 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 11421 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) / (𝐺‘𝑘)) = ((𝐹‘𝑘) · (1 / (𝐺‘𝑘)))) |
29 | 20 | eqcomd 2829 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) = ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘)) |
30 | 29 | oveq2d 7174 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) · (1 / (𝐺‘𝑘))) = ((𝐹‘𝑘) · ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘))) |
31 | 27, 28, 30 | 3eqtrd 2862 | . . 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 41892 | . 2 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · (1 / 𝐵))) |
33 | climcl 14858 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
34 | 7, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
35 | climcl 14858 | . . . 4 ⊢ (𝐺 ⇝ 𝐵 → 𝐵 ∈ ℂ) | |
36 | 10, 35 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
37 | 34, 36, 11 | divrecd 11421 | . 2 ⊢ (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
38 | 32, 37 | breqtrrd 5096 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 ≠ wne 3018 Vcvv 3496 ∖ cdif 3935 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 · cmul 10544 / cdiv 11299 ℤcz 11984 ℤ≥cuz 12246 ⇝ cli 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 |
This theorem is referenced by: stirlinglem8 42373 fourierdlem103 42501 fourierdlem104 42502 |
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