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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 28668 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 NrmCVeccnv 28361 CHilOLDchlo 28662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-cbn 28640 df-hlo 28663 |
This theorem is referenced by: htthlem 28694 axhfvadd-zf 28759 axhvcom-zf 28760 axhvass-zf 28761 axhvaddid-zf 28763 axhfvmul-zf 28764 axhvmulid-zf 28765 axhvmulass-zf 28766 axhvdistr1-zf 28767 axhvdistr2-zf 28768 axhvmul0-zf 28769 axhis2-zf 28772 axhis3-zf 28773 axhcompl-zf 28775 hilcompl 28978 |
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