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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 30910 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑈 ∈ NrmCVec |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 NrmCVeccnv 30603 CHilOLDchlo 30904 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-cbn 30882 df-hlo 30905 |
| This theorem is referenced by: htthlem 30936 axhfvadd-zf 31001 axhvcom-zf 31002 axhvass-zf 31003 axhvaddid-zf 31005 axhfvmul-zf 31006 axhvmulid-zf 31007 axhvmulass-zf 31008 axhvdistr1-zf 31009 axhvdistr2-zf 31010 axhvmul0-zf 31011 axhis2-zf 31014 axhis3-zf 31015 axhcompl-zf 31017 hilcompl 31220 |
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