Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 28674 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 NrmCVeccnv 28367 CHilOLDchlo 28668 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-cbn 28646 df-hlo 28669 |
This theorem is referenced by: htthlem 28700 axhfvadd-zf 28765 axhvcom-zf 28766 axhvass-zf 28767 axhvaddid-zf 28769 axhfvmul-zf 28770 axhvmulid-zf 28771 axhvmulass-zf 28772 axhvdistr1-zf 28773 axhvdistr2-zf 28774 axhvmul0-zf 28775 axhis2-zf 28778 axhis3-zf 28779 axhcompl-zf 28781 hilcompl 28984 |
Copyright terms: Public domain | W3C validator |