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Mirrors > Home > ILE Home > Th. List > climconst | 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 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | uzid 9471 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
4 | climconst.1 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrrdi 2258 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | 5 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑀 ∈ 𝑍) |
7 | climconst.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
8 | 7 | subidd 8188 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
9 | 8 | fveq2d 5484 | . . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = (abs‘0)) |
10 | abs0 10986 | . . . . . . . 8 ⊢ (abs‘0) = 0 | |
11 | 9, 10 | eqtrdi 2213 | . . . . . . 7 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = 0) |
12 | 11 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) = 0) |
13 | rpgt0 9592 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
14 | 13 | adantl 275 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
15 | 12, 14 | eqbrtrd 3998 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) < 𝑥) |
16 | 15 | ralrimivw 2538 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) |
17 | fveq2 5480 | . . . . . . 7 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀)) | |
18 | 17, 4 | eqtr4di 2215 | . . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = 𝑍) |
19 | 18 | raleqdv 2665 | . . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥)) |
20 | 19 | rspcev 2825 | . . . 4 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
21 | 6, 16, 20 | syl2anc 409 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
22 | 21 | ralrimiva 2537 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
23 | climconst.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
24 | climconst.5 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
25 | 7 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
26 | 4, 1, 23, 24, 7, 25 | clim2c 11211 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥)) |
27 | 22, 26 | mpbird 166 | 1 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 0cc0 7744 < clt 7924 − cmin 8060 ℤcz 9182 ℤ≥cuz 9457 ℝ+crp 9580 abscabs 10925 ⇝ cli 11205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-rsqrt 10926 df-abs 10927 df-clim 11206 |
This theorem is referenced by: climconst2 11218 fsum3cvg 11305 fproddccvg 11499 fprodntrivap 11511 |
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