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Theorem nvsass 24020
Description: Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvscl.1  |-  X  =  ( BaseSet `  U )
nvscl.4  |-  S  =  ( .sOLD `  U )
Assertion
Ref Expression
nvsass  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )

Proof of Theorem nvsass
StepHypRef Expression
1 eqid 2443 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 24005 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2443 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 23993 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 23995 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 23994 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vcass 23944 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
102, 9sylan 471 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5430  (class class class)co 6103   1stc1st 6587   CCcc 9292    x. cmul 9299   CVecOLDcvc 23935   NrmCVeccnv 23974   +vcpv 23975   BaseSetcba 23976   .sOLDcns 23977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-1st 6589  df-2nd 6590  df-vc 23936  df-nv 23982  df-va 23985  df-ba 23986  df-sm 23987  df-0v 23988  df-nmcv 23990
This theorem is referenced by:  nvscom  24021  nvmul0or  24044  nvnncan  24055  nvpi  24066  smcnlem  24104  ipval3  24116  ipdirilem  24241  ipasslem2  24244  ipasslem4  24246  ipasslem5  24247  ipasslem10  24251  ipasslem11  24252  minvecolem2  24288  hlmulass  24319
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