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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 30128 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | |
2 | bnnv 30106 | . 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 29824 CBanccbn 30102 CHilOLDchlo 30125 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-cbn 30103 df-hlo 30126 |
This theorem is referenced by: hlnvi 30132 hlvc 30133 hladdf 30139 hlcom 30140 hlass 30141 hl0cl 30142 hladdid 30143 hlmulf 30144 hlmulid 30145 hlmulass 30146 hldi 30147 hldir 30148 hlmul0 30149 hlipf 30150 hlipcj 30151 hlipgt0 30154 |
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