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Mirrors > Home > MPE Home > Th. List > climabs | Structured version Visualization version GIF version |
Description: Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.) |
Ref | Expression |
---|---|
climcn1lem.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climcn1lem.2 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climcn1lem.4 | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
climcn1lem.5 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climcn1lem.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
climabs.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘))) |
Ref | Expression |
---|---|
climabs | ⊢ (𝜑 → 𝐺 ⇝ (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climcn1lem.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climcn1lem.2 | . 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
3 | climcn1lem.4 | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
4 | climcn1lem.5 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | climcn1lem.6 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
6 | absf 14558 | . . 3 ⊢ abs:ℂ⟶ℝ | |
7 | ax-resscn 10392 | . . 3 ⊢ ℝ ⊆ ℂ | |
8 | fss 6357 | . . 3 ⊢ ((abs:ℂ⟶ℝ ∧ ℝ ⊆ ℂ) → abs:ℂ⟶ℂ) | |
9 | 6, 7, 8 | mp2an 679 | . 2 ⊢ abs:ℂ⟶ℂ |
10 | abscn2 14816 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((abs‘𝑧) − (abs‘𝐴))) < 𝑥)) | |
11 | climabs.7 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘))) | |
12 | 1, 2, 3, 4, 5, 9, 10, 11 | climcn1lem 14820 | 1 ⊢ (𝜑 → 𝐺 ⇝ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3830 class class class wbr 4929 ⟶wf 6184 ‘cfv 6188 ℂcc 10333 ℝcr 10334 ℤcz 11793 ℤ≥cuz 12058 abscabs 14454 ⇝ cli 14702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-sup 8701 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-seq 13185 df-exp 13245 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 |
This theorem is referenced by: iserabs 15030 ulmdvlem1 24691 dchrisumlem3 25769 dvgrat 40057 binomcxplemrat 40095 |
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