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Theorem hvadd32 24371
Description: Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvadd32  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( A  +h  C )  +h  B ) )

Proof of Theorem hvadd32
StepHypRef Expression
1 ax-hvcom 24338 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  =  ( C  +h  B ) )
21oveq2d 6106 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( A  +h  ( C  +h  B
) ) )
323adant1 1001 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( A  +h  ( C  +h  B ) ) )
4 ax-hvass 24339 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
5 ax-hvass 24339 . . 3  |-  ( ( A  e.  ~H  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
( A  +h  C
)  +h  B )  =  ( A  +h  ( C  +h  B
) ) )
653com23 1188 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  C
)  +h  B )  =  ( A  +h  ( C  +h  B
) ) )
73, 4, 63eqtr4d 2483 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( A  +h  C )  +h  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761  (class class class)co 6090   ~Hchil 24256    +h cva 24257
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 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-hvcom 24338  ax-hvass 24339
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-iota 5378  df-fv 5423  df-ov 6093
This theorem is referenced by:  hvadd4  24373  hvadd32i  24391
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