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Theorem hldir 22410
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  =  ( .s OLD `  U
)
Assertion
Ref Expression
hldir  |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )

Proof of Theorem hldir
StepHypRef Expression
1 hlnv 22393 . 2  |-  ( U  e.  CHil OLD  ->  U  e.  NrmCVec )
2 hldi.1 . . 3  |-  X  =  ( BaseSet `  U )
3 hldi.2 . . 3  |-  G  =  ( +v `  U
)
4 hldi.4 . . 3  |-  S  =  ( .s OLD `  U
)
52, 3, 4nvdir 22112 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  -> 
( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
61, 5sylan 458 1  |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988    + caddc 8993   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s
OLDcns 22066   CHil
OLDchlo 22387
This theorem is referenced by:  axhvdistr2-zf  22494
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-1st 6349  df-2nd 6350  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079  df-cbn 22365  df-hlo 22388
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