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Theorem nvdir 25820
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 2402 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25802 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 nvdi.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 25790 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvdi.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 25792 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvdi.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 25791 . . 3  |-  X  =  ran  G
94, 6, 8vcdir 25740 . 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 469 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5525  (class class class)co 6234   1stc1st 6736   CCcc 9440    + caddc 9445   CVecOLDcvc 25732   NrmCVeccnv 25771   +vcpv 25772   BaseSetcba 25773   .sOLDcns 25774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-1st 6738  df-2nd 6739  df-vc 25733  df-nv 25779  df-va 25782  df-ba 25783  df-sm 25784  df-0v 25785  df-nmcv 25787
This theorem is referenced by:  nvge0  25871  smcnlem  25901  ipidsq  25917  ip2i  26037  ipasslem1  26040  ipasslem11  26049  hldir  26118
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