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Mirrors > Home > MPE Home > Th. List > divcnvshft | Structured version Visualization version GIF version |
Description: Limit of a ratio function. (Contributed by Scott Fenton, 16-Dec-2017.) |
Ref | Expression |
---|---|
divcnvshft.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
divcnvshft.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
divcnvshft.3 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcnvshft.4 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
divcnvshft.5 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
divcnvshft.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐴 / (𝑘 + 𝐵))) |
Ref | Expression |
---|---|
divcnvshft | ⊢ (𝜑 → 𝐹 ⇝ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcnvshft.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcnv 15193 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) |
4 | nnssz 11989 | . . . . . . 7 ⊢ ℕ ⊆ ℤ | |
5 | resmpt 5891 | . . . . . . 7 ⊢ (ℕ ⊆ ℤ → ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ ℕ) = (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚))) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ ℕ) = (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) |
7 | nnuz 12268 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
8 | 7 | reseq2i 5836 | . . . . . 6 ⊢ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ ℕ) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
9 | 6, 8 | eqtr3i 2846 | . . . . 5 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
10 | 9 | breq1i 5059 | . . . 4 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0) |
11 | 1z 11999 | . . . . 5 ⊢ 1 ∈ ℤ | |
12 | zex 11977 | . . . . . 6 ⊢ ℤ ∈ V | |
13 | 12 | mptex 6972 | . . . . 5 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ∈ V |
14 | climres 14917 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ∈ V) → (((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0)) | |
15 | 11, 13, 14 | mp2an 690 | . . . 4 ⊢ (((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
16 | 10, 15 | bitri 277 | . . 3 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
17 | 3, 16 | sylib 220 | . 2 ⊢ (𝜑 → (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
18 | divcnvshft.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
19 | divcnvshft.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
20 | divcnvshft.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
21 | divcnvshft.5 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
22 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ∈ V) |
23 | uzssz 12251 | . . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
24 | 18, 23 | eqsstri 3989 | . . . . . . . 8 ⊢ 𝑍 ⊆ ℤ |
25 | 24 | sseli 3951 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
26 | 25 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ) |
27 | 20 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℤ) |
28 | 26, 27 | zaddcld 12078 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐵) ∈ ℤ) |
29 | oveq2 7150 | . . . . . 6 ⊢ (𝑚 = (𝑘 + 𝐵) → (𝐴 / 𝑚) = (𝐴 / (𝑘 + 𝐵))) | |
30 | eqid 2821 | . . . . . 6 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) = (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) | |
31 | ovex 7175 | . . . . . 6 ⊢ (𝐴 / (𝑘 + 𝐵)) ∈ V | |
32 | 29, 30, 31 | fvmpt 6754 | . . . . 5 ⊢ ((𝑘 + 𝐵) ∈ ℤ → ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚))‘(𝑘 + 𝐵)) = (𝐴 / (𝑘 + 𝐵))) |
33 | 28, 32 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚))‘(𝑘 + 𝐵)) = (𝐴 / (𝑘 + 𝐵))) |
34 | divcnvshft.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐴 / (𝑘 + 𝐵))) | |
35 | 33, 34 | eqtr4d 2859 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚))‘(𝑘 + 𝐵)) = (𝐹‘𝑘)) |
36 | 18, 19, 20, 21, 22, 35 | climshft2 14924 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0)) |
37 | 17, 36 | mpbird 259 | 1 ⊢ (𝜑 → 𝐹 ⇝ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ⊆ wss 3924 class class class wbr 5052 ↦ cmpt 5132 ↾ cres 5543 ‘cfv 6341 (class class class)co 7142 ℂcc 10521 0cc0 10523 1c1 10524 + caddc 10526 / cdiv 11283 ℕcn 11624 ℤcz 11968 ℤ≥cuz 12230 ⇝ cli 14826 |
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 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-n0 11885 df-z 11969 df-uz 12231 df-rp 12377 df-fl 13152 df-seq 13360 df-exp 13420 df-shft 14411 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-clim 14830 df-rlim 14831 |
This theorem is referenced by: trireciplem 15202 lgamcvg2 25618 binomcxplemrat 40772 |
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