Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hlnv | Structured version Visualization version GIF version |
Description: Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlnv | ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlobn 28592 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
2 | bnnv 28570 | . 2 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 NrmCVeccnv 28288 CBanccbn 28566 CHilOLDchlo 28589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-cbn 28567 df-hlo 28590 |
This theorem is referenced by: hlnvi 28596 hlvc 28597 hladdf 28603 hlcom 28604 hlass 28605 hl0cl 28606 hladdid 28607 hlmulf 28608 hlmulid 28609 hlmulass 28610 hldi 28611 hldir 28612 hlmul0 28613 hlipf 28614 hlipcj 28615 hlipgt0 28618 |
Copyright terms: Public domain | W3C validator |