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Theorem nvscom 22102
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  =  ( .s OLD `  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 9068 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6088 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) S C )  =  ( ( B  x.  A ) S C ) )
323adant3 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )  ->  (
( A  x.  B
) S C )  =  ( ( B  x.  A ) S C ) )
43adantl 453 . 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  =  ( .s OLD `  U
)
75, 6nvsass 22101 . 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 943 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )  <->  ( B  e.  CC  /\  A  e.  CC  /\  C  e.  X ) )
95, 6nvsass 22101 . . 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 462 . 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 2475 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980    x. cmul 8987   NrmCVeccnv 22055   BaseSetcba 22057   .s OLDcns 22058
This theorem is referenced by:  nvmdi  22123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-mulcom 9046
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
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