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Mirrors > Home > MPE Home > Th. List > climconst | Structured version Visualization version GIF version |
Description: An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
climconst.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climconst.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climconst.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
climconst.4 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
climconst.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
Ref | Expression |
---|---|
climconst | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climconst.2 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | uzid 12261 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
4 | climconst.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrrdi 2926 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | climconst.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
7 | 6 | subidd 10987 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
8 | 7 | fveq2d 6676 | . . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = (abs‘0)) |
9 | abs0 14647 | . . . . . . . 8 ⊢ (abs‘0) = 0 | |
10 | 8, 9 | syl6eq 2874 | . . . . . . 7 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = 0) |
11 | 10 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) = 0) |
12 | rpgt0 12404 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
13 | 12 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
14 | 11, 13 | eqbrtrd 5090 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) < 𝑥) |
15 | 14 | ralrimivw 3185 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) |
16 | fveq2 6672 | . . . . . . 7 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀)) | |
17 | 16, 4 | syl6eqr 2876 | . . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = 𝑍) |
18 | 17 | raleqdv 3417 | . . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥)) |
19 | 18 | rspcev 3625 | . . . 4 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
20 | 5, 15, 19 | syl2an2r 683 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
21 | 20 | ralrimiva 3184 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
22 | climconst.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
23 | climconst.5 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
24 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
25 | 4, 1, 22, 23, 6, 24 | clim2c 14864 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥)) |
26 | 21, 25 | mpbird 259 | 1 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 < clt 10677 − cmin 10872 ℤcz 11984 ℤ≥cuz 12246 ℝ+crp 12392 abscabs 14595 ⇝ cli 14843 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 |
This theorem is referenced by: climconst2 14907 fsumcvg 15071 expcnv 15221 ntrivcvgfvn0 15257 fprodcvg 15286 fprodntriv 15298 faclim2 32982 clim1fr1 41889 climneg 41898 ioodvbdlimc1lem2 42224 ioodvbdlimc2lem 42226 fourierdlem103 42501 fourierdlem104 42502 etransclem48 42574 |
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