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Theorem nvscom 21203
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 8839 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 5889 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) S C )  =  ( ( B  x.  A ) S C ) )
323adant3 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )  ->  (
( A  x.  B
) S C )  =  ( ( B  x.  A ) S C ) )
43adantl 452 . 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 21202 . 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 941 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )  <->  ( B  e.  CC  /\  A  e.  CC  /\  C  e.  X ) )
95, 6nvsass 21202 . . 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 461 . 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 2336 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   NrmCVeccnv 21156   BaseSetcba 21158   .s OLDcns 21159
This theorem is referenced by:  nvmdi  21224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-mulcom 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172
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