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Mirrors > Home > HSE Home > Th. List > chsssh | Structured version Visualization version GIF version |
Description: Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chsssh | ⊢ Cℋ ⊆ Sℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsh 31052 | . 2 ⊢ (𝑥 ∈ Cℋ → 𝑥 ∈ Sℋ ) | |
2 | 1 | ssriv 3984 | 1 ⊢ Cℋ ⊆ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3947 Sℋ csh 30756 Cℋ cch 30757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-xp 5686 df-cnv 5688 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fv 6559 df-ov 7427 df-ch 31049 |
This theorem is referenced by: chex 31054 chsspwh 31075 chintcli 31159 shatomistici 32189 |
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