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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 30916 | . 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 2106 CPreHilOLDccphlo 30841 CBanccbn 30891 CHilOLDchlo 30914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-in 3970 df-hlo 30915 |
This theorem is referenced by: hlpar2 30925 hlpar 30926 hlipdir 30941 hlipass 30942 htthlem 30946 |
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