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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 14424 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) |
4 | nnssz 11274 | . . . . . . 7 ⊢ ℕ ⊆ ℤ | |
5 | resmpt 5369 | . . . . . . 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 11599 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
8 | 7 | reseq2i 5314 | . . . . . 6 ⊢ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ ℕ) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
9 | 6, 8 | eqtr3i 2634 | . . . . 5 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
10 | 9 | breq1i 4590 | . . . 4 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0) |
11 | 1z 11284 | . . . . 5 ⊢ 1 ∈ ℤ | |
12 | zex 11263 | . . . . . 6 ⊢ ℤ ∈ V | |
13 | 12 | mptex 6390 | . . . . 5 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ∈ V |
14 | climres 14154 | . . . . 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 704 | . . . 4 ⊢ (((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
16 | 10, 15 | bitri 263 | . . 3 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
17 | 3, 16 | sylib 207 | . 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 11583 | . . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
24 | 18, 23 | eqsstri 3598 | . . . . . . . 8 ⊢ 𝑍 ⊆ ℤ |
25 | 24 | sseli 3564 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
26 | 25 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ) |
27 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℤ) |
28 | 26, 27 | zaddcld 11362 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐵) ∈ ℤ) |
29 | oveq2 6557 | . . . . . 6 ⊢ (𝑚 = (𝑘 + 𝐵) → (𝐴 / 𝑚) = (𝐴 / (𝑘 + 𝐵))) | |
30 | eqid 2610 | . . . . . 6 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) = (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) | |
31 | ovex 6577 | . . . . . 6 ⊢ (𝐴 / (𝑘 + 𝐵)) ∈ V | |
32 | 29, 30, 31 | fvmpt 6191 | . . . . 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 2647 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚))‘(𝑘 + 𝐵)) = (𝐹‘𝑘)) |
36 | 18, 19, 20, 21, 22, 35 | climshft2 14161 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0)) |
37 | 17, 36 | mpbird 246 | 1 ⊢ (𝜑 → 𝐹 ⇝ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 / cdiv 10563 ℕcn 10897 ℤcz 11254 ℤ≥cuz 11563 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-seq 12664 df-exp 12723 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 |
This theorem is referenced by: binomcxplemrat 37571 |
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