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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 28693 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ) | |
2 | 1 | sselda 3895 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2083 ℋchba 28383 Cℋ cch 28393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 ax-sep 5101 ax-hilex 28463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-xp 5456 df-cnv 5458 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fv 6240 df-ov 7026 df-sh 28671 df-ch 28685 |
This theorem is referenced by: pjhtheu2 28880 pjspansn 29041 pjid 29159 atom1d 29817 sumdmdii 29879 |
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