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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 29537 | . 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) | |
2 | 1 | simprbi 497 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 CPreHilOLDccphlo 29462 CBanccbn 29512 CHilOLDchlo 29535 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-in 3905 df-hlo 29536 |
This theorem is referenced by: hlpar2 29546 hlpar 29547 hlipdir 29562 hlipass 29563 htthlem 29567 |
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