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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 6887 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
4 | 2 | fmpttd 6872 | . . 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 14937 | . 2 ⊢ (𝜑 → (𝐹 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
9 | 3, 8 | eqbrtrrd 5081 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ⇝𝑟 (𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2108 ∀wral 3136 ∃wrex 3137 class class class wbr 5057 ↦ cmpt 5137 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 < clt 10667 − cmin 10862 ℝ+crp 12381 abscabs 14585 ⇝𝑟 crli 14834 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-pm 8401 df-rlim 14838 |
This theorem is referenced by: rlimabs 14957 rlimcj 14958 rlimre 14959 rlimim 14960 |
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