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Mirrors > Home > MPE Home > Th. List > rlimcn1b | Structured version Visualization version GIF version |
Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.) |
Ref | Expression |
---|---|
rlimcn1b.1 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑋) |
rlimcn1b.2 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
rlimcn1b.3 | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶) |
rlimcn1b.4 | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
rlimcn1b.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥)) |
Ref | Expression |
---|---|
rlimcn1b | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimcn1b.4 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
2 | rlimcn1b.1 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑋) | |
3 | 1, 2 | cofmpt 7148 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
4 | 2 | fmpttd 7131 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑋) |
5 | rlimcn1b.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
6 | rlimcn1b.3 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶) | |
7 | rlimcn1b.5 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥)) | |
8 | 4, 5, 6, 1, 7 | rlimcn1 15592 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
9 | 3, 8 | eqbrtrrd 5179 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 class class class wbr 5155 ↦ cmpt 5238 ∘ ccom 5688 ⟶wf 6552 ‘cfv 6556 (class class class)co 7426 ℂcc 11158 < clt 11300 − cmin 11496 ℝ+crp 13030 abscabs 15241 ⇝𝑟 crli 15489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-pm 8860 df-rlim 15493 |
This theorem is referenced by: rlimabs 15613 rlimcj 15614 rlimre 15615 rlimim 15616 |
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