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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 7152 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
4 | 2 | fmpttd 7135 | . . 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 15621 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
9 | 3, 8 | eqbrtrrd 5172 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 < clt 11293 − cmin 11490 ℝ+crp 13032 abscabs 15270 ⇝𝑟 crli 15518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8868 df-rlim 15522 |
This theorem is referenced by: rlimabs 15642 rlimcj 15643 rlimre 15644 rlimim 15645 |
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