Theorem ch0 5133

Description: The zero vector belongs to any closed subspace of a Hilbert space.

Assertion

Ref	Expression
ch0	⊢ (H ∈ Cℋ → 0v ∈ H)

Proof of Theorem ch0

Step	Hyp	Ref	Expression
1		chsh 5131	. 2 ⊢ (H ∈ Cℋ → H ∈ Sℋ )
2		sh0 5122	. 2 ⊢ (H ∈ Sℋ → 0v ∈ H)
3	1, 2	syl 12	1 ⊢ (H ∈ Cℋ → 0v ∈ H)

Syntax hints:   → wi 2   ∈ wcel 1092  0_vc0v 4961   S_ℋ csh 4967   C_ℋ cch 4968

This theorem is referenced by:  projlem8 5200  projlem16 5208  projlem20 5212  pjthlem14 5238  pjth 5239  omlsi 5250  strlem1 5691

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sh 5114  df-ch 5127

metamath.org