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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 29492 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ) | |
2 | 1 | sselda 3917 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ℋchba 29182 Cℋ cch 29192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 ax-sep 5218 ax-hilex 29262 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-xp 5586 df-cnv 5588 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fv 6426 df-ov 7258 df-sh 29470 df-ch 29484 |
This theorem is referenced by: pjhtheu2 29679 pjspansn 29840 pjid 29958 atom1d 30616 sumdmdii 30678 |
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