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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 6664 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
4 | 2 | fmpttd 6649 | . . 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 14727 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
9 | 3, 8 | eqbrtrrd 4910 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 class class class wbr 4886 ↦ cmpt 4965 ∘ ccom 5359 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 < clt 10411 − cmin 10606 ℝ+crp 12137 abscabs 14381 ⇝𝑟 crli 14624 |
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 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 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-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-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-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-pm 8143 df-rlim 14628 |
This theorem is referenced by: rlimabs 14747 rlimcj 14748 rlimre 14749 rlimim 14750 |
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