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Theorem hvadd32 24571
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 24538 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  =  ( C  +h  B ) )
21oveq2d 6206 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( A  +h  ( C  +h  B
) ) )
323adant1 1006 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( A  +h  ( C  +h  B ) ) )
4 ax-hvass 24539 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
5 ax-hvass 24539 . . 3  |-  ( ( A  e.  ~H  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
( A  +h  C
)  +h  B )  =  ( A  +h  ( C  +h  B
) ) )
653com23 1194 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  C
)  +h  B )  =  ( A  +h  ( C  +h  B
) ) )
73, 4, 63eqtr4d 2502 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 965    = wceq 1370    e. wcel 1758  (class class class)co 6190   ~Hchil 24456    +h cva 24457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-hvcom 24538  ax-hvass 24539
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-iota 5479  df-fv 5524  df-ov 6193
This theorem is referenced by:  hvadd4  24573  hvadd32i  24591
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