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Theorem hvadd4i 21637
Description: Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvass.1  |-  A  e. 
~H
hvass.2  |-  B  e. 
~H
hvass.3  |-  C  e. 
~H
hvadd4.4  |-  D  e. 
~H
Assertion
Ref Expression
hvadd4i  |-  ( ( A  +h  B )  +h  ( C  +h  D ) )  =  ( ( A  +h  C )  +h  ( B  +h  D ) )

Proof of Theorem hvadd4i
StepHypRef Expression
1 hvass.1 . 2  |-  A  e. 
~H
2 hvass.2 . 2  |-  B  e. 
~H
3 hvass.3 . 2  |-  C  e. 
~H
4 hvadd4.4 . 2  |-  D  e. 
~H
5 hvadd4 21615 . 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 ) ) )
61, 2, 3, 4, 5mp4an 654 1  |-  ( ( A  +h  B )  +h  ( C  +h  D ) )  =  ( ( A  +h  C )  +h  ( B  +h  D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   ~Hchil 21499    +h cva 21500
This theorem is referenced by:  hvsubsub4i  21638  hvsubcan2i  21643  pjaddii  22254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861
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