![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9518 | . . . . . . 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 2271 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | 5 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑀 ∈ 𝑍) |
7 | climconst.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
8 | 7 | subidd 8233 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
9 | 8 | fveq2d 5514 | . . . . . . . 8 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = (abs‘0)) |
10 | abs0 11038 | . . . . . . . 8 ⊢ (abs‘0) = 0 | |
11 | 9, 10 | eqtrdi 2226 | . . . . . . 7 ⊢ (𝜑 → (abs‘(𝐴 − 𝐴)) = 0) |
12 | 11 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) = 0) |
13 | rpgt0 9639 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
14 | 13 | adantl 277 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
15 | 12, 14 | eqbrtrd 4022 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (abs‘(𝐴 − 𝐴)) < 𝑥) |
16 | 15 | ralrimivw 2551 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥) |
17 | fveq2 5510 | . . . . . . 7 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀)) | |
18 | 17, 4 | eqtr4di 2228 | . . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = 𝑍) |
19 | 18 | raleqdv 2678 | . . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐴 − 𝐴)) < 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐴 − 𝐴)) < 𝑥)) |
20 | 19 | rspcev 2841 | . . . 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 2550 | . 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 11263 | . 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 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 ℂcc 7787 0cc0 7789 < clt 7969 − cmin 8105 ℤcz 9229 ℤ≥cuz 9504 ℝ+crp 9627 abscabs 10977 ⇝ cli 11257 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-2 8954 df-n0 9153 df-z 9230 df-uz 9505 df-rp 9628 df-seqfrec 10419 df-exp 10493 df-cj 10822 df-rsqrt 10978 df-abs 10979 df-clim 11258 |
This theorem is referenced by: climconst2 11270 fsum3cvg 11357 fproddccvg 11551 fprodntrivap 11563 |
Copyright terms: Public domain | W3C validator |