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Axiom ax-hvass 22458
Description: Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-hvass  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )

Detailed syntax breakdown of Axiom ax-hvass
StepHypRef Expression
1 cA . . . 4  class  A
2 chil 22375 . . . 4  class  ~H
31, 2wcel 1721 . . 3  wff  A  e. 
~H
4 cB . . . 4  class  B
54, 2wcel 1721 . . 3  wff  B  e. 
~H
6 cC . . . 4  class  C
76, 2wcel 1721 . . 3  wff  C  e. 
~H
83, 5, 7w3a 936 . 2  wff  ( A  e.  ~H  /\  B  e.  ~H  /\  C  e. 
~H )
9 cva 22376 . . . . 5  class  +h
101, 4, 9co 6040 . . . 4  class  ( A  +h  B )
1110, 6, 9co 6040 . . 3  class  ( ( A  +h  B )  +h  C )
124, 6, 9co 6040 . . . 4  class  ( B  +h  C )
131, 12, 9co 6040 . . 3  class  ( A  +h  ( B  +h  C ) )
1411, 13wceq 1649 . 2  wff  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) )
158, 14wi 4 1  wff  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
Colors of variables: wff set class
This axiom is referenced by:  hvadd32  22489  hvadd12  22490  hvadd4  22491  hvpncan  22494  hvaddsubass  22496  hvassi  22508  hilablo  22615  spanunsni  23034  hoaddassi  23232
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