Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hlcms | Structured version Visualization version GIF version |
Description: Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
Ref | Expression |
---|---|
hlcms | ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlbn 23159 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) | |
2 | bncms 23141 | . 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 CMetSpccms 23129 Bancbn 23130 ℂHilchl 23131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-bn 23133 df-hl 23134 |
This theorem is referenced by: pjthlem2 23209 |
Copyright terms: Public domain | W3C validator |