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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 15829 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) |
4 | nnssz 12608 | . . . . . . 7 ⊢ ℕ ⊆ ℤ | |
5 | resmpt 6036 | . . . . . . 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 12893 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
8 | 7 | reseq2i 5976 | . . . . . 6 ⊢ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ ℕ) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
9 | 6, 8 | eqtr3i 2755 | . . . . 5 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
10 | 9 | breq1i 5150 | . . . 4 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0) |
11 | 1z 12620 | . . . . 5 ⊢ 1 ∈ ℤ | |
12 | zex 12595 | . . . . . 6 ⊢ ℤ ∈ V | |
13 | 12 | mptex 7230 | . . . . 5 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ∈ V |
14 | climres 15549 | . . . . 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 274 | . . 3 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
17 | 3, 16 | sylib 217 | . 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 12871 | . . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
24 | 18, 23 | eqsstri 4007 | . . . . . . . 8 ⊢ 𝑍 ⊆ ℤ |
25 | 24 | sseli 3968 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
26 | 25 | adantl 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ) |
27 | 20 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℤ) |
28 | 26, 27 | zaddcld 12698 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐵) ∈ ℤ) |
29 | oveq2 7423 | . . . . . 6 ⊢ (𝑚 = (𝑘 + 𝐵) → (𝐴 / 𝑚) = (𝐴 / (𝑘 + 𝐵))) | |
30 | eqid 2725 | . . . . . 6 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) = (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) | |
31 | ovex 7448 | . . . . . 6 ⊢ (𝐴 / (𝑘 + 𝐵)) ∈ V | |
32 | 29, 30, 31 | fvmpt 6999 | . . . . 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 2768 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚))‘(𝑘 + 𝐵)) = (𝐹‘𝑘)) |
36 | 18, 19, 20, 21, 22, 35 | climshft2 15556 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0)) |
37 | 17, 36 | mpbird 256 | 1 ⊢ (𝜑 → 𝐹 ⇝ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3940 class class class wbr 5143 ↦ cmpt 5226 ↾ cres 5674 ‘cfv 6542 (class class class)co 7415 ℂcc 11134 0cc0 11136 1c1 11137 + caddc 11139 / cdiv 11899 ℕcn 12240 ℤcz 12586 ℤ≥cuz 12850 ⇝ cli 15458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-fl 13787 df-seq 13997 df-exp 14057 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-rlim 15463 |
This theorem is referenced by: trireciplem 15838 lgamcvg2 27003 binomcxplemrat 43851 |
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