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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 15450 | . . 3 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | climcn1lem.7 | . . . 4 ⊢ 𝐻:ℂ⟶ℂ | |
7 | 6 | ffvelcdmi 7085 | . . 3 ⊢ (𝑧 ∈ ℂ → (𝐻‘𝑧) ∈ ℂ) |
8 | 7 | adantl 481 | . 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 579 | . 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 15543 | 1 ⊢ (𝜑 → 𝐺 ⇝ (𝐻‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 < clt 11255 − cmin 11451 ℤcz 12565 ℤ≥cuz 12829 ℝ+crp 12981 abscabs 15188 ⇝ cli 15435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-pre-lttri 11190 ax-pre-lttrn 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-neg 11454 df-z 12566 df-uz 12830 df-clim 15439 |
This theorem is referenced by: climabs 15555 climcj 15556 climre 15557 climim 15558 sinccvglem 35122 |
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