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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 30819 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 NrmCVeccnv 30512 CHilOLDchlo 30813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-iota 6496 df-fv 6552 df-cbn 30791 df-hlo 30814 |
This theorem is referenced by: htthlem 30845 axhfvadd-zf 30910 axhvcom-zf 30911 axhvass-zf 30912 axhvaddid-zf 30914 axhfvmul-zf 30915 axhvmulid-zf 30916 axhvmulass-zf 30917 axhvdistr1-zf 30918 axhvdistr2-zf 30919 axhvmul0-zf 30920 axhis2-zf 30923 axhis3-zf 30924 axhcompl-zf 30926 hilcompl 31129 |
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