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Mirrors > Home > MPE Home > Th. List > ishl | Structured version Visualization version GIF version |
Description: The predicate "is a subcomplex Hilbert space". A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
ishl | ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hl 24704 | . 2 ⊢ ℂHil = (Ban ∩ ℂPreHil) | |
2 | 1 | elin2 4158 | 1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ℂPreHilccph 24533 Bancbn 24700 ℂHilchl 24701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3448 df-in 3918 df-hl 24704 |
This theorem is referenced by: hlbn 24730 hlcph 24731 ishl2 24737 cphssphl 24738 cmslsschl 24744 chlcsschl 24745 |
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