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Mirrors > Home > MPE Home > Th. List > climre | Structured version Visualization version GIF version |
Description: Limit of the real part 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) |
climre.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (ℜ‘(𝐹‘𝑘))) |
Ref | Expression |
---|---|
climre | ⊢ (𝜑 → 𝐺 ⇝ (ℜ‘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 | ref 14561 | . . 3 ⊢ ℜ:ℂ⟶ℝ | |
7 | ax-resscn 10672 | . . 3 ⊢ ℝ ⊆ ℂ | |
8 | fss 6521 | . . 3 ⊢ ((ℜ:ℂ⟶ℝ ∧ ℝ ⊆ ℂ) → ℜ:ℂ⟶ℂ) | |
9 | 6, 7, 8 | mp2an 692 | . 2 ⊢ ℜ:ℂ⟶ℂ |
10 | recn2 15048 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((ℜ‘𝑧) − (ℜ‘𝐴))) < 𝑥)) | |
11 | climre.7 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (ℜ‘(𝐹‘𝑘))) | |
12 | 1, 2, 3, 4, 5, 9, 10, 11 | climcn1lem 15050 | 1 ⊢ (𝜑 → 𝐺 ⇝ (ℜ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 class class class wbr 5030 ⟶wf 6335 ‘cfv 6339 ℂcc 10613 ℝcr 10614 ℤcz 12062 ℤ≥cuz 12324 ℜcre 14546 ⇝ cli 14931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 |
This theorem is referenced by: mbflim 24420 |
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