Theorem chss 28426

Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
chss	⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ)

Proof of Theorem chss

Step	Hyp	Ref	Expression
1		chsh 28421	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ )
2		shss 28407	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
3	1, 2	syl 17	1 ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2145   ⊆ wss 3723   ℋchil 28116   S_ℋ csh 28125   C_ℋ cch 28126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-hilex 28196

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fv 6038  df-ov 6799  df-sh 28404  df-ch 28418

This theorem is referenced by:  chel  28427  pjhcl  28600  dfch2  28606  shlub  28613  chsscon2  28701  chscllem2  28837  pjvec  28895  pjocvec  28896  pjhf  28907  elpjrn  29389