HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvadd12 Structured version   Unicode version

Theorem hvadd12 26366
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 26332 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  =  ( B  +h  A ) )
21oveq1d 6293 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  +h  C
)  =  ( ( B  +h  A )  +h  C ) )
323adant3 1017 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( B  +h  A )  +h  C ) )
4 ax-hvass 26333 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
5 ax-hvass 26333 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  ~H  /\  C  e.  ~H )  ->  (
( B  +h  A
)  +h  C )  =  ( B  +h  ( A  +h  C
) ) )
653com12 1201 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( B  +h  A
)  +h  C )  =  ( B  +h  ( A  +h  C
) ) )
73, 4, 63eqtr3d 2451 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842  (class class class)co 6278   ~Hchil 26250    +h cva 26251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-hvcom 26332  ax-hvass 26333
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-ov 6281
This theorem is referenced by:  hvaddsub12  26369
  Copyright terms: Public domain W3C validator