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Theorem nvdir 25199
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
nvdir  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )

Proof of Theorem nvdir
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25181 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 nvdi.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 25169 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvdi.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 25171 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvdi.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 25170 . . 3  |-  X  =  ran  G
94, 6, 8vcdir 25119 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
102, 9sylan 471 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   1stc1st 6779   CCcc 9486    + caddc 9491   CVecOLDcvc 25111   NrmCVeccnv 25150   +vcpv 25151   BaseSetcba 25152   .sOLDcns 25153
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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-reu 2821  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-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-1st 6781  df-2nd 6782  df-vc 25112  df-nv 25158  df-va 25161  df-ba 25162  df-sm 25163  df-0v 25164  df-nmcv 25166
This theorem is referenced by:  nvge0  25250  smcnlem  25280  ipidsq  25296  ip2i  25416  ipasslem1  25419  ipasslem11  25428  hldir  25497
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