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Mirrors > Home > MPE Home > Th. List > climcn1lem | Structured version Visualization version GIF version |
Description: The limit of a continuous function, theorem form. (Contributed 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) |
climcn1lem.7 | ⊢ 𝐻:ℂ⟶ℂ |
climcn1lem.8 | ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐻‘𝑧) − (𝐻‘𝐴))) < 𝑥)) |
climcn1lem.9 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐻‘(𝐹‘𝑘))) |
Ref | Expression |
---|---|
climcn1lem | ⊢ (𝜑 → 𝐺 ⇝ (𝐻‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climcn1lem.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climcn1lem.5 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climcn1lem.2 | . . 3 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
4 | climcl 14848 | . . 3 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | climcn1lem.7 | . . . 4 ⊢ 𝐻:ℂ⟶ℂ | |
7 | 6 | ffvelrni 6827 | . . 3 ⊢ (𝑧 ∈ ℂ → (𝐻‘𝑧) ∈ ℂ) |
8 | 7 | adantl 485 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐻‘𝑧) ∈ ℂ) |
9 | climcn1lem.4 | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
10 | climcn1lem.8 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐻‘𝑧) − (𝐻‘𝐴))) < 𝑥)) | |
11 | 5, 10 | sylan 583 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐻‘𝑧) − (𝐻‘𝐴))) < 𝑥)) |
12 | climcn1lem.6 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
13 | climcn1lem.9 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐻‘(𝐹‘𝑘))) | |
14 | 1, 2, 5, 8, 3, 9, 11, 12, 13 | climcn1 14940 | 1 ⊢ (𝜑 → 𝐺 ⇝ (𝐻‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 < clt 10664 − cmin 10859 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14585 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-clim 14837 |
This theorem is referenced by: climabs 14952 climcj 14953 climre 14954 climim 14955 sinccvglem 33028 |
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