Theorem hvaddcl 4999

Description: Closure of vector addition.

Hypotheses

Ref	Expression
hvaddcl.1	|- A e. H~
hvaddcl.2	|- B e. H~

Assertion

Ref	Expression
hvaddcl	|- (A +v B) e. H~

Proof of Theorem hvaddcl

Step	Hyp	Ref	Expression
1		hvaddcl.1	. 2 |- A e. H~
2		hvaddcl.2	. 2 |- B e. H~
3		ax-hvaddcl 4984	. 2 |- ((A e. H~ /\ B e. H~) -> (A +v B) e. H~)
4	1, 2, 3	mp2an 520	1 |- (A +v B) e. H~

Syntax hints:   e. wcel 1092  (class class class)co 3001  H~chil 4958   +v cva 4959

This theorem is referenced by:  hvsubass 5027  hvsubsub4 5031  hv2times 5033  hvsubadd 5038  normlem0 5062  normlem9 5070  norm-ii 5086  normpyth 5090  norm3dif 5094  normpar 5099  normpar2 5100  projlem5 5197  projlem7 5199  projlem18 5210  pjthlem1 5225  pjcomp 5565  pjadd 5566

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-hvaddcl 4984

This theorem depends on definitions:  df-bi 128  df-an 198

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