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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 12891 | . . . . . 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 2850 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | climconst.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
7 | 6 | subidd 11606 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
8 | 7 | fveq2d 6911 | . . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = (abs‘0)) |
9 | abs0 15321 | . . . . . . . 8 ⊢ (abs‘0) = 0 | |
10 | 8, 9 | eqtrdi 2791 | . . . . . . 7 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = 0) |
11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) = 0) |
12 | rpgt0 13045 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
13 | 12 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
14 | 11, 13 | eqbrtrd 5170 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) < 𝑥) |
15 | 14 | ralrimivw 3148 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) |
16 | fveq2 6907 | . . . . . . 7 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀)) | |
17 | 16, 4 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = 𝑍) |
18 | 17 | raleqdv 3324 | . . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥)) |
19 | 18 | rspcev 3622 | . . . 4 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
20 | 5, 15, 19 | syl2an2r 685 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥) |
21 | 20 | ralrimiva 3144 | . 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 480 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
25 | 4, 1, 22, 23, 6, 24 | clim2c 15538 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥)) |
26 | 21, 25 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 < clt 11293 − cmin 11490 ℤcz 12611 ℤ≥cuz 12876 ℝ+crp 13032 abscabs 15270 ⇝ cli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 |
This theorem is referenced by: climconst2 15581 fsumcvg 15745 expcnv 15897 ntrivcvgfvn0 15932 fprodcvg 15963 fprodntriv 15975 faclim2 35728 clim1fr1 45557 climneg 45566 ioodvbdlimc1lem2 45888 ioodvbdlimc2lem 45890 fourierdlem103 46165 fourierdlem104 46166 etransclem48 46238 |
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