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Theorem nvscom 21951
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 9002 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6028 . . . 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 21950 . 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 21950 . . 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 2420 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 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   CCcc 8914    x. cmul 8921   NrmCVeccnv 21904   BaseSetcba 21906   .s OLDcns 21907
This theorem is referenced by:  nvmdi  21972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-mulcom 8980
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-1st 6281  df-2nd 6282  df-vc 21866  df-nv 21912  df-va 21915  df-ba 21916  df-sm 21917  df-0v 21918  df-nmcv 21920
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