| 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 5250 | . . 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 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
| 14 | 12 | eldifad 3963 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
| 15 | eldifsni 4790 | . . . . . . 7 ⊢ ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) → (𝐺‘𝑘) ≠ 0) | |
| 16 | 12, 15 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≠ 0) |
| 17 | 14, 16 | reccld 12036 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) ∈ ℂ) |
| 18 | eqid 2737 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) | |
| 19 | 18 | fvmpt2 7027 | . . . . 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 6920 | . . . . . 6 ⊢ 𝑍 ∈ V |
| 22 | 21 | mptex 7243 | . . . . 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 45624 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) ⇝ (1 / 𝐵)) |
| 25 | climdivf.11 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
| 26 | 20, 17 | eqeltrd 2841 | . . 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 12046 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) / (𝐺‘𝑘)) = ((𝐹‘𝑘) · (1 / (𝐺‘𝑘)))) |
| 29 | 20 | eqcomd 2743 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) = ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘)) |
| 30 | 29 | oveq2d 7447 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) · (1 / (𝐺‘𝑘))) = ((𝐹‘𝑘) · ((𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘)))‘𝑘))) |
| 31 | 27, 28, 30 | 3eqtrd 2781 | . . 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 45619 | . 2 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · (1 / 𝐵))) |
| 33 | climcl 15535 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
| 34 | 7, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 35 | climcl 15535 | . . . 4 ⊢ (𝐺 ⇝ 𝐵 → 𝐵 ∈ ℂ) | |
| 36 | 10, 35 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 37 | 34, 36, 11 | divrecd 12046 | . 2 ⊢ (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
| 38 | 32, 37 | breqtrrd 5171 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 / cdiv 11920 ℤcz 12613 ℤ≥cuz 12878 ⇝ 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-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-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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 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-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 |
| This theorem is referenced by: stirlinglem8 46096 fourierdlem103 46224 fourierdlem104 46225 |
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