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Theorem hvadd4 24436
Description: Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvadd4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  +h  ( C  +h  D
) )  =  ( ( A  +h  C
)  +h  ( B  +h  D ) ) )

Proof of Theorem hvadd4
StepHypRef Expression
1 hvadd32 24434 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( A  +h  C )  +h  B ) )
21oveq1d 6104 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( A  +h  B )  +h  C
)  +h  D )  =  ( ( ( A  +h  C )  +h  B )  +h  D ) )
323expa 1187 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( ( A  +h  B )  +h  C )  +h  D )  =  ( ( ( A  +h  C )  +h  B
)  +h  D ) )
43adantrr 716 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  +h  B
)  +h  C )  +h  D )  =  ( ( ( A  +h  C )  +h  B )  +h  D
) )
5 hvaddcl 24412 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
6 ax-hvass 24402 . . . 4  |-  ( ( ( A  +h  B
)  e.  ~H  /\  C  e.  ~H  /\  D  e.  ~H )  ->  (
( ( A  +h  B )  +h  C
)  +h  D )  =  ( ( A  +h  B )  +h  ( C  +h  D
) ) )
763expb 1188 . . 3  |-  ( ( ( A  +h  B
)  e.  ~H  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  +h  B
)  +h  C )  +h  D )  =  ( ( A  +h  B )  +h  ( C  +h  D ) ) )
85, 7sylan 471 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  +h  B
)  +h  C )  +h  D )  =  ( ( A  +h  B )  +h  ( C  +h  D ) ) )
9 hvaddcl 24412 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  C
)  e.  ~H )
10 ax-hvass 24402 . . . . 5  |-  ( ( ( A  +h  C
)  e.  ~H  /\  B  e.  ~H  /\  D  e.  ~H )  ->  (
( ( A  +h  C )  +h  B
)  +h  D )  =  ( ( A  +h  C )  +h  ( B  +h  D
) ) )
11103expb 1188 . . . 4  |-  ( ( ( A  +h  C
)  e.  ~H  /\  ( B  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  +h  C
)  +h  B )  +h  D )  =  ( ( A  +h  C )  +h  ( B  +h  D ) ) )
129, 11sylan 471 . . 3  |-  ( ( ( A  e.  ~H  /\  C  e.  ~H )  /\  ( B  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  +h  C
)  +h  B )  +h  D )  =  ( ( A  +h  C )  +h  ( B  +h  D ) ) )
1312an4s 822 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  +h  C
)  +h  B )  +h  D )  =  ( ( A  +h  C )  +h  ( B  +h  D ) ) )
144, 8, 133eqtr3d 2481 1  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  +h  B )  +h  ( C  +h  D
) )  =  ( ( A  +h  C
)  +h  ( B  +h  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756  (class class class)co 6089   ~Hchil 24319    +h cva 24320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-hfvadd 24400  ax-hvcom 24401  ax-hvass 24402
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092
This theorem is referenced by:  hvsub4  24437  hvadd4i  24458  shscli  24718  spanunsni  24980  mayete3i  25129  mayete3iOLD  25130  lnophsi  25403
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