Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 28591 | . 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 28516 CBanccbn 28566 CHilOLDchlo 28589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-hlo 28590 |
This theorem is referenced by: hlpar2 28600 hlpar 28601 hlipdir 28616 hlipass 28617 htthlem 28621 |
Copyright terms: Public domain | W3C validator |