![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 15805 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0) |
4 | nnssz 12584 | . . . . . . 7 ⊢ ℕ ⊆ ℤ | |
5 | resmpt 6031 | . . . . . . 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 12869 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
8 | 7 | reseq2i 5972 | . . . . . 6 ⊢ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ ℕ) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
9 | 6, 8 | eqtr3i 2756 | . . . . 5 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) = ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) |
10 | 9 | breq1i 5148 | . . . 4 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 / 𝑚)) ⇝ 0 ↔ ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0) |
11 | 1z 12596 | . . . . 5 ⊢ 1 ∈ ℤ | |
12 | zex 12571 | . . . . . 6 ⊢ ℤ ∈ V | |
13 | 12 | mptex 7220 | . . . . 5 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ∈ V |
14 | climres 15525 | . . . . 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 689 | . . . 4 ⊢ (((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ↾ (ℤ≥‘1)) ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0) |
16 | 10, 15 | bitri 275 | . . 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 12847 | . . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
24 | 18, 23 | eqsstri 4011 | . . . . . . . 8 ⊢ 𝑍 ⊆ ℤ |
25 | 24 | sseli 3973 | . . . . . . 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 12674 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐵) ∈ ℤ) |
29 | oveq2 7413 | . . . . . 6 ⊢ (𝑚 = (𝑘 + 𝐵) → (𝐴 / 𝑚) = (𝐴 / (𝑘 + 𝐵))) | |
30 | eqid 2726 | . . . . . 6 ⊢ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) = (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) | |
31 | ovex 7438 | . . . . . 6 ⊢ (𝐴 / (𝑘 + 𝐵)) ∈ V | |
32 | 29, 30, 31 | fvmpt 6992 | . . . . 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 2769 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ ℤ ↦ (𝐴 / 𝑚))‘(𝑘 + 𝐵)) = (𝐹‘𝑘)) |
36 | 18, 19, 20, 21, 22, 35 | climshft2 15532 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ (𝑚 ∈ ℤ ↦ (𝐴 / 𝑚)) ⇝ 0)) |
37 | 17, 36 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 ⇝ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 class class class wbr 5141 ↦ cmpt 5224 ↾ cres 5671 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 / cdiv 11875 ℕcn 12216 ℤcz 12562 ℤ≥cuz 12826 ⇝ cli 15434 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fl 13763 df-seq 13973 df-exp 14033 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 |
This theorem is referenced by: trireciplem 15814 lgamcvg2 26942 binomcxplemrat 43690 |
Copyright terms: Public domain | W3C validator |