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| Mirrors > Home > MPE Home > Th. List > phnvi | Structured version Visualization version GIF version | ||
| Description: Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| phnvi.1 | ⊢ 𝑈 ∈ CPreHilOLD |
| Ref | Expression |
|---|---|
| phnvi | ⊢ 𝑈 ∈ NrmCVec |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | phnvi.1 | . 2 ⊢ 𝑈 ∈ CPreHilOLD | |
| 2 | phnv 30833 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑈 ∈ NrmCVec |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 NrmCVeccnv 30603 CPreHilOLDccphlo 30831 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-ss 3968 df-ph 30832 |
| This theorem is referenced by: elimph 30839 ip0i 30844 ip1ilem 30845 ip2i 30847 ipdirilem 30848 ipasslem1 30850 ipasslem2 30851 ipasslem4 30853 ipasslem5 30854 ipasslem7 30855 ipasslem8 30856 ipasslem9 30857 ipasslem10 30858 ipasslem11 30859 ip2dii 30863 pythi 30869 siilem1 30870 siilem2 30871 siii 30872 ipblnfi 30874 ip2eqi 30875 ajfuni 30878 |
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