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Theorem hvadd4 25645
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 25643 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( A  +h  C )  +h  B ) )
21oveq1d 6298 . . . 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 1196 . . 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 25621 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
6 ax-hvass 25611 . . . 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 1197 . . 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 25621 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  C
)  e.  ~H )
10 ax-hvass 25611 . . . . 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 1197 . . . 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 824 . 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 2516 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 973    = wceq 1379    e. wcel 1767  (class class class)co 6283   ~Hchil 25528    +h cva 25529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-hfvadd 25609  ax-hvcom 25610  ax-hvass 25611
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-fv 5595  df-ov 6286
This theorem is referenced by:  hvsub4  25646  hvadd4i  25667  shscli  25927  spanunsni  26189  mayete3i  26338  mayete3iOLD  26339  lnophsi  26612
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