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Mirrors > Home > HSE Home > Th. List > chel | Structured version Visualization version GIF version |
Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chel | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chss 31162 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ) | |
2 | 1 | sselda 3979 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ℋchba 30852 Cℋ cch 30862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-hilex 30932 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fv 6562 df-ov 7427 df-sh 31140 df-ch 31154 |
This theorem is referenced by: pjhtheu2 31349 pjspansn 31510 pjid 31628 atom1d 32286 sumdmdii 32348 |
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