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Theorem hvadd32 22489
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 22457 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  =  ( C  +h  B ) )
21oveq2d 6056 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( A  +h  ( C  +h  B
) ) )
323adant1 975 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  C ) )  =  ( A  +h  ( C  +h  B ) ) )
4 ax-hvass 22458 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
5 ax-hvass 22458 . . 3  |-  ( ( A  e.  ~H  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
( A  +h  C
)  +h  B )  =  ( A  +h  ( C  +h  B
) ) )
653com23 1159 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  C
)  +h  B )  =  ( A  +h  ( C  +h  B
) ) )
73, 4, 63eqtr4d 2446 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( A  +h  C )  +h  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721  (class class class)co 6040   ~Hchil 22375    +h cva 22376
This theorem is referenced by:  hvadd4  22491  hvadd32i  22509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-hvcom 22457  ax-hvass 22458
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043
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