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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 28671 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
2 | bnnv 28649 | . 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 2111 NrmCVeccnv 28367 CBanccbn 28645 CHilOLDchlo 28668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-cbn 28646 df-hlo 28669 |
This theorem is referenced by: hlnvi 28675 hlvc 28676 hladdf 28682 hlcom 28683 hlass 28684 hl0cl 28685 hladdid 28686 hlmulf 28687 hlmulid 28688 hlmulass 28689 hldi 28690 hldir 28691 hlmul0 28692 hlipf 28693 hlipcj 28694 hlipgt0 28697 |
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