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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 30917 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
2 | bnnv 30895 | . 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 2106 NrmCVeccnv 30613 CBanccbn 30891 CHilOLDchlo 30914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-cbn 30892 df-hlo 30915 |
This theorem is referenced by: hlnvi 30921 hlvc 30922 hladdf 30928 hlcom 30929 hlass 30930 hl0cl 30931 hladdid 30932 hlmulf 30933 hlmulid 30934 hlmulass 30935 hldi 30936 hldir 30937 hlmul0 30938 hlipf 30939 hlipcj 30940 hlipgt0 30943 |
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