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| Mirrors > Home > MPE Home > Th. List > hlph | Structured version Visualization version GIF version | ||
| Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hlph | ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ishlo 30906 | . 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) | |
| 2 | 1 | simprbi 496 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 CPreHilOLDccphlo 30831 CBanccbn 30881 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-hlo 30905 | 
| This theorem is referenced by: hlpar2 30915 hlpar 30916 hlipdir 30931 hlipass 30932 htthlem 30936 | 
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