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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 14638 | . . 3 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | climcn1lem.7 | . . . 4 ⊢ 𝐻:ℂ⟶ℂ | |
7 | 6 | ffvelrni 6622 | . . 3 ⊢ (𝑧 ∈ ℂ → (𝐻‘𝑧) ∈ ℂ) |
8 | 7 | adantl 475 | . 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 575 | . 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 14730 | 1 ⊢ (𝜑 → 𝐺 ⇝ (𝐻‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 class class class wbr 4886 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 < clt 10411 − cmin 10606 ℤcz 11728 ℤ≥cuz 11992 ℝ+crp 12137 abscabs 14381 ⇝ cli 14623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-neg 10609 df-z 11729 df-uz 11993 df-clim 14627 |
This theorem is referenced by: climabs 14742 climcj 14743 climre 14744 climim 14745 sinccvglem 32163 |
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