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Mirrors > Home > MPE Home > Th. List > hlnvi | Structured version Visualization version GIF version |
Description: Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlnvi.1 | ⊢ 𝑈 ∈ CHilOLD |
Ref | Expression |
---|---|
hlnvi | ⊢ 𝑈 ∈ NrmCVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnvi.1 | . 2 ⊢ 𝑈 ∈ CHilOLD | |
2 | hlnv 28595 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 NrmCVeccnv 28288 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: htthlem 28621 axhfvadd-zf 28686 axhvcom-zf 28687 axhvass-zf 28688 axhvaddid-zf 28690 axhfvmul-zf 28691 axhvmulid-zf 28692 axhvmulass-zf 28693 axhvdistr1-zf 28694 axhvdistr2-zf 28695 axhvmul0-zf 28696 axhis2-zf 28699 axhis3-zf 28700 axhcompl-zf 28702 hilcompl 28905 |
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