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Theorem hvadd4 22539
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 22537 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( A  +h  C )  +h  B ) )
21oveq1d 6097 . . . 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 1154 . . 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 699 . 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 22516 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
6 ax-hvass 22506 . . . 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 1155 . . 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 459 . 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 22516 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  C
)  e.  ~H )
10 ax-hvass 22506 . . . . 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 1155 . . . 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 459 . . 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 801 . 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 2477 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726  (class class class)co 6082   ~Hchil 22423    +h cva 22424
This theorem is referenced by:  hvsub4  22540  hvadd4i  22561  shscli  22820  spanunsni  23082  mayete3i  23231  mayete3iOLD  23232  lnophsi  23505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-hfvadd 22504  ax-hvcom 22505  ax-hvass 22506
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085
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