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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 23943 | . 2 ⊢ ℂHil = (Ban ∩ ℂPreHil) | |
2 | 1 | elin2 4177 | 1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ℂPreHilccph 23773 Bancbn 23939 ℂHilchl 23940 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-in 3946 df-hl 23943 |
This theorem is referenced by: hlbn 23969 hlcph 23970 ishl2 23976 cphssphl 23977 cmslsschl 23983 chlcsschl 23984 |
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