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Theorem nvscom 25355
Description: Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvscl.1  |-  X  =  ( BaseSet `  U )
nvscl.4  |-  S  =  ( .sOLD `  U )
Assertion
Ref Expression
nvscom  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( A S ( B S C ) )  =  ( B S ( A S C ) ) )

Proof of Theorem nvscom
StepHypRef Expression
1 mulcom 9590 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6310 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) S C )  =  ( ( B  x.  A ) S C ) )
323adant3 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )  ->  (
( A  x.  B
) S C )  =  ( ( B  x.  A ) S C ) )
43adantl 466 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( A  x.  B ) S C )  =  ( ( B  x.  A ) S C ) )
5 nvscl.1 . . 3  |-  X  =  ( BaseSet `  U )
6 nvscl.4 . . 3  |-  S  =  ( .sOLD `  U )
75, 6nvsass 25354 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
8 3ancoma 980 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )  <->  ( B  e.  CC  /\  A  e.  CC  /\  C  e.  X ) )
95, 6nvsass 25354 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( B  e.  CC  /\  A  e.  CC  /\  C  e.  X ) )  -> 
( ( B  x.  A ) S C )  =  ( B S ( A S C ) ) )
108, 9sylan2b 475 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( B  x.  A ) S C )  =  ( B S ( A S C ) ) )
114, 7, 103eqtr3d 2516 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( A S ( B S C ) )  =  ( B S ( A S C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   CCcc 9502    x. cmul 9509   NrmCVeccnv 25308   BaseSetcba 25310   .sOLDcns 25311
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-mulcom 9568
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-1st 6795  df-2nd 6796  df-vc 25270  df-nv 25316  df-va 25319  df-ba 25320  df-sm 25321  df-0v 25322  df-nmcv 25324
This theorem is referenced by:  nvmdi  25376
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