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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 28272 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 NrmCVeccnv 27964 CHilOLDchlo 28266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-iota 6064 df-fv 6109 df-cbn 28244 df-hlo 28267 |
This theorem is referenced by: htthlem 28299 axhfvadd-zf 28364 axhvcom-zf 28365 axhvass-zf 28366 axhvaddid-zf 28368 axhfvmul-zf 28369 axhvmulid-zf 28370 axhvmulass-zf 28371 axhvdistr1-zf 28372 axhvdistr2-zf 28373 axhvmul0-zf 28374 axhis2-zf 28377 axhis3-zf 28378 axhcompl-zf 28380 hilcompl 28583 |
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