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Mirrors > Home > MPE Home > Th. List > elcncf1ii | Structured version Visualization version GIF version |
Description: Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.) |
Ref | Expression |
---|---|
elcncf1i.1 | ⊢ 𝐹:𝐴⟶𝐵 |
elcncf1i.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+) |
elcncf1i.3 | ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥 − 𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)) |
Ref | Expression |
---|---|
elcncf1ii | ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elcncf1i.1 | . . . 4 ⊢ 𝐹:𝐴⟶𝐵 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐹:𝐴⟶𝐵) |
3 | elcncf1i.2 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)) |
5 | elcncf1i.3 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥 − 𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥 − 𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))) |
7 | 2, 4, 6 | elcncf1di 22506 | . 2 ⊢ (⊤ → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵))) |
8 | 7 | trud 1484 | 1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ⊤wtru 1476 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 < clt 9953 − cmin 10145 ℝ+crp 11708 abscabs 13822 –cn→ccncf 22487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-cncf 22489 |
This theorem is referenced by: logcnlem5 24192 |
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