Theorem ch0 9098

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

Assertion

Ref	Expression
ch0	|- (H e. CH -> 0h e. H)

Proof of Theorem ch0

Step	Hyp	Ref	Expression
1		chsh 9096	. 2 |- (H e. CH -> H e. SH)
2		sh0 9084	. 2 |- (H e. SH -> 0h e. H)
3	1, 2	syl 10	1 |- (H e. CH -> 0h e. H)

Syntax hints:   -> wi 3   e. wcel 958  0hc0v 8791  SHcsh 8797  CHcch 8798

This theorem is referenced by:  projlem8 9193  projlem16 9201  projlem20 9205  pjthlem14 9232  pjth 9233  omlsi 9245  nonbool 9596  strlem1 10177

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-sh 9076  df-ch 9092

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