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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 5256 | . . 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 3975 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
15 | eldifsni 4795 | . . . . . . 7 ⊢ ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) → (𝐺‘𝑘) ≠ 0) | |
16 | 12, 15 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≠ 0) |
17 | 14, 16 | reccld 12034 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 / (𝐺‘𝑘)) ∈ ℂ) |
18 | eqid 2735 | . . . . . 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 6921 | . . . . . 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 45565 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (1 / (𝐺‘𝑘))) ⇝ (1 / 𝐵)) |
25 | climdivf.11 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
26 | 20, 17 | eqeltrd 2839 | . . 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 12044 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) / (𝐺‘𝑘)) = ((𝐹‘𝑘) · (1 / (𝐺‘𝑘)))) |
29 | 20 | eqcomd 2741 | . . . . 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 2779 | . . 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 45560 | . 2 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · (1 / 𝐵))) |
33 | climcl 15532 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
34 | 7, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
35 | climcl 15532 | . . . 4 ⊢ (𝐺 ⇝ 𝐵 → 𝐵 ∈ ℂ) | |
36 | 10, 35 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
37 | 34, 36, 11 | divrecd 12044 | . 2 ⊢ (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
38 | 32, 37 | breqtrrd 5176 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 {csn 4631 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 · cmul 11158 / cdiv 11918 ℤcz 12611 ℤ≥cuz 12876 ⇝ cli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 |
This theorem is referenced by: stirlinglem8 46037 fourierdlem103 46165 fourierdlem104 46166 |
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