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Theorem hvadd12 25614
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 25580 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  =  ( B  +h  A ) )
21oveq1d 6290 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  +h  C
)  =  ( ( B  +h  A )  +h  C ) )
323adant3 1011 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( B  +h  A )  +h  C ) )
4 ax-hvass 25581 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
5 ax-hvass 25581 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  ~H  /\  C  e.  ~H )  ->  (
( B  +h  A
)  +h  C )  =  ( B  +h  ( A  +h  C
) ) )
653com12 1195 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( B  +h  A
)  +h  C )  =  ( B  +h  ( A  +h  C
) ) )
73, 4, 63eqtr3d 2509 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762  (class class class)co 6275   ~Hchil 25498    +h cva 25499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-hvcom 25580  ax-hvass 25581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-ov 6278
This theorem is referenced by:  hvaddsub12  25617
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