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Theorem hvadd12 22538
Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvadd12  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) ) )

Proof of Theorem hvadd12
StepHypRef Expression
1 ax-hvcom 22505 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  =  ( B  +h  A ) )
21oveq1d 6097 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  +h  C
)  =  ( ( B  +h  A )  +h  C ) )
323adant3 978 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( B  +h  A )  +h  C ) )
4 ax-hvass 22506 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
5 ax-hvass 22506 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  ~H  /\  C  e.  ~H )  ->  (
( B  +h  A
)  +h  C )  =  ( B  +h  ( A  +h  C
) ) )
653com12 1158 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( B  +h  A
)  +h  C )  =  ( B  +h  ( A  +h  C
) ) )
73, 4, 63eqtr3d 2477 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726  (class class class)co 6082   ~Hchil 22423    +h cva 22424
This theorem is referenced by:  hvaddsub12  22541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-hvcom 22505  ax-hvass 22506
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085
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