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Theorem nvdi 25187
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  =  ( .sOLD `  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 2460 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25170 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 nvdi.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 25158 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvdi.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 25160 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvdi.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 25159 . . 3  |-  X  =  ran  G
94, 6, 8vcdi 25107 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
102, 9sylan 471 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   1stc1st 6772   CCcc 9479   CVecOLDcvc 25100   NrmCVeccnv 25139   +vcpv 25140   BaseSetcba 25141   .sOLDcns 25142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-1st 6774  df-2nd 6775  df-vc 25101  df-nv 25147  df-va 25150  df-ba 25151  df-sm 25152  df-0v 25153  df-nmcv 25155
This theorem is referenced by:  nvmdi  25207  nvaddsub4  25218  nvnncan  25220  nvdif  25230  nvpi  25231  ipdirilem  25406  hldi  25485
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