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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 14673 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) |
4 | nnssz 11478 | . . . . . . 7 ⊢ ℕ ⊆ ℤ | |
5 | resmpt 5527 | . . . . . . 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 11805 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
8 | 7 | reseq2i 5468 | . . . . . 6 ⊢ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ ℕ) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
9 | 6, 8 | eqtr3i 2716 | . . . . 5 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
10 | 9 | breq1i 4735 | . . . 4 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0) |
11 | 1z 11488 | . . . . 5 ⊢ 1 ∈ ℤ | |
12 | zex 11467 | . . . . . 6 ⊢ ℤ ∈ V | |
13 | 12 | mptex 6570 | . . . . 5 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ∈ V |
14 | climres 14394 | . . . . 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 710 | . . . 4 ⊢ (((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
16 | 10, 15 | bitri 264 | . . 3 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
17 | 3, 16 | sylib 208 | . 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 11788 | . . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
24 | 18, 23 | eqsstri 3709 | . . . . . . . 8 ⊢ 𝑍 ⊆ ℤ |
25 | 24 | sseli 3673 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
26 | 25 | adantl 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ) |
27 | 20 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℤ) |
28 | 26, 27 | zaddcld 11567 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐵) ∈ ℤ) |
29 | oveq2 6741 | . . . . . 6 ⊢ (𝑚 = (𝑘 + 𝐵) → (𝐴 / 𝑚) = (𝐴 / (𝑘 + 𝐵))) | |
30 | eqid 2692 | . . . . . 6 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) = (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) | |
31 | ovex 6761 | . . . . . 6 ⊢ (𝐴 / (𝑘 + 𝐵)) ∈ V | |
32 | 29, 30, 31 | fvmpt 6364 | . . . . 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 2729 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚))‘(𝑘 + 𝐵)) = (𝐹‘𝑘)) |
36 | 18, 19, 20, 21, 22, 35 | climshft2 14401 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0)) |
37 | 17, 36 | mpbird 247 | 1 ⊢ (𝜑 → 𝐹 ⇝ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1564 ∈ wcel 2071 Vcvv 3272 ⊆ wss 3648 class class class wbr 4728 ↦ cmpt 4805 ↾ cres 5188 ‘cfv 5969 (class class class)co 6733 ℂcc 10015 0cc0 10017 1c1 10018 + caddc 10020 / cdiv 10765 ℕcn 11101 ℤcz 11458 ℤ≥cuz 11768 ⇝ cli 14303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 ax-pre-sup 10095 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-iun 4598 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-om 7151 df-2nd 7254 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-er 7830 df-pm 7945 df-en 8041 df-dom 8042 df-sdom 8043 df-sup 8432 df-inf 8433 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-div 10766 df-nn 11102 df-2 11160 df-3 11161 df-n0 11374 df-z 11459 df-uz 11769 df-rp 11915 df-fl 12676 df-seq 12885 df-exp 12944 df-shft 13895 df-cj 13927 df-re 13928 df-im 13929 df-sqrt 14063 df-abs 14064 df-clim 14307 df-rlim 14308 |
This theorem is referenced by: binomcxplemrat 38936 |
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