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Theorem hldi 25965
Description: Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hldi.1  |-  X  =  ( BaseSet `  U )
hldi.2  |-  G  =  ( +v `  U
)
hldi.4  |-  S  =  ( .sOLD `  U )
Assertion
Ref Expression
hldi  |-  ( ( U  e.  CHilOLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )

Proof of Theorem hldi
StepHypRef Expression
1 hlnv 25949 . 2  |-  ( U  e.  CHilOLD  ->  U  e.  NrmCVec )
2 hldi.1 . . 3  |-  X  =  ( BaseSet `  U )
3 hldi.2 . . 3  |-  G  =  ( +v `  U
)
4 hldi.4 . . 3  |-  S  =  ( .sOLD `  U )
52, 3, 4nvdi 25667 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
61, 5sylan 469 1  |-  ( ( U  e.  CHilOLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   ` cfv 5513  (class class class)co 6218   CCcc 9423   NrmCVeccnv 25619   +vcpv 25620   BaseSetcba 25621   .sOLDcns 25622   CHilOLDchlo 25943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-1st 6721  df-2nd 6722  df-vc 25581  df-nv 25627  df-va 25630  df-ba 25631  df-sm 25632  df-0v 25633  df-nmcv 25635  df-cbn 25921  df-hlo 25944
This theorem is referenced by:  axhvdistr1-zf  26049
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