Theorem hvaddcl 4999

Description: Closure of vector addition.

Hypotheses

Ref	Expression
hvaddcl.1	⊢ A ∈ ℋ
hvaddcl.2	⊢ B ∈ ℋ

Assertion

Ref	Expression
hvaddcl	⊢ (A +v B) ∈ ℋ

Proof of Theorem hvaddcl

Step	Hyp	Ref	Expression
1		hvaddcl.1	. 2 ⊢ A ∈ ℋ
2		hvaddcl.2	. 2 ⊢ B ∈ ℋ
3		ax-hvaddcl 4984	. 2 ⊢ ((A ∈ ℋ ∧ B ∈ ℋ ) → (A +v B) ∈ ℋ )
4	1, 2, 3	mp2an 520	1 ⊢ (A +v B) ∈ ℋ

Syntax hints:   ∈ 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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