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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 9555 | . . . . . . 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 2281 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | 5 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑀 ∈ 𝑍) |
7 | climconst.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
8 | 7 | subidd 8269 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
9 | 8 | fveq2d 5531 | . . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = (abs‘0)) |
10 | abs0 11080 | . . . . . . . 8 ⊢ (abs‘0) = 0 | |
11 | 9, 10 | eqtrdi 2236 | . . . . . . 7 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = 0) |
12 | 11 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) = 0) |
13 | rpgt0 9678 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
14 | 13 | adantl 277 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
15 | 12, 14 | eqbrtrd 4037 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) < 𝑥) |
16 | 15 | ralrimivw 2561 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) |
17 | fveq2 5527 | . . . . . . 7 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀)) | |
18 | 17, 4 | eqtr4di 2238 | . . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = 𝑍) |
19 | 18 | raleqdv 2689 | . . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥)) |
20 | 19 | rspcev 2853 | . . . 4 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
21 | 6, 16, 20 | syl2anc 411 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
22 | 21 | ralrimiva 2560 | . 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 276 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
26 | 4, 1, 23, 24, 7, 25 | clim2c 11305 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥)) |
27 | 22, 26 | mpbird 167 | 1 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ∃wrex 2466 class class class wbr 4015 ‘cfv 5228 (class class class)co 5888 ℂcc 7822 0cc0 7824 < clt 8005 − cmin 8141 ℤcz 9266 ℤ≥cuz 9541 ℝ+crp 9666 abscabs 11019 ⇝ cli 11299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-n0 9190 df-z 9267 df-uz 9542 df-rp 9667 df-seqfrec 10459 df-exp 10533 df-cj 10864 df-rsqrt 11020 df-abs 11021 df-clim 11300 |
This theorem is referenced by: climconst2 11312 fsum3cvg 11399 fproddccvg 11593 fprodntrivap 11605 |
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