MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvscom Structured version   Unicode version

Theorem nvscom 24009
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 9368 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6106 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) S C )  =  ( ( B  x.  A ) S C ) )
323adant3 1008 . . 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 24008 . 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 972 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )  <->  ( B  e.  CC  /\  A  e.  CC  /\  C  e.  X ) )
95, 6nvsass 24008 . . 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 2483 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 965    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   CCcc 9280    x. cmul 9287   NrmCVeccnv 23962   BaseSetcba 23964   .sOLDcns 23965
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-mulcom 9346
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-1st 6577  df-2nd 6578  df-vc 23924  df-nv 23970  df-va 23973  df-ba 23974  df-sm 23975  df-0v 23976  df-nmcv 23978
This theorem is referenced by:  nvmdi  24030
  Copyright terms: Public domain W3C validator