Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hlbn | Structured version Visualization version GIF version |
Description: Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
Ref | Expression |
---|---|
hlbn | ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishl 24114 | . 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ℂPreHilccph 23918 Bancbn 24085 ℂHilchl 24086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-in 3850 df-hl 24089 |
This theorem is referenced by: hlcms 24118 hlprlem 24119 cmslsschl 24129 chlcsschl 24130 |
Copyright terms: Public domain | W3C validator |