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Theorem nvdi 22116
Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvdi.1  |-  X  =  ( BaseSet `  U )
nvdi.2  |-  G  =  ( +v `  U
)
nvdi.4  |-  S  =  ( .s OLD `  U
)
Assertion
Ref Expression
nvdi  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )

Proof of Theorem nvdi
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 22099 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 nvdi.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 22087 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvdi.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 22089 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvdi.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 22088 . . 3  |-  X  =  ran  G
94, 6, 8vcdi 22036 . 2  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
102, 9sylan 459 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   1stc1st 6350   CCcc 8993   CVec OLDcvc 22029   NrmCVeccnv 22068   +vcpv 22069   BaseSetcba 22070   .s
OLDcns 22071
This theorem is referenced by:  nvmdi  22136  nvaddsub4  22147  nvnncan  22149  nvdif  22159  nvpi  22160  ipdirilem  22335  hldi  22414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-1st 6352  df-2nd 6353  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-nmcv 22084
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