Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23505 | . 2 ⊢ (⊤ → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵))) |
8 | 7 | mptru 1544 | 1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ⊤wtru 1538 ∈ wcel 2114 ⊆ wss 3938 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 < clt 10677 − cmin 10872 ℝ+crp 12392 abscabs 14595 –cn→ccncf 23486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-cncf 23488 |
This theorem is referenced by: logcnlem5 25231 |
Copyright terms: Public domain | W3C validator |