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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 31253 | . 2 ⊢ (𝑥 ∈ Cℋ → 𝑥 ∈ Sℋ ) | |
2 | 1 | ssriv 3999 | 1 ⊢ Cℋ ⊆ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3963 Sℋ csh 30957 Cℋ cch 30958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fv 6571 df-ov 7434 df-ch 31250 |
This theorem is referenced by: chex 31255 chsspwh 31276 chintcli 31360 shatomistici 32390 |
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