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Theorem hldir 24454
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
hldir  |-  ( ( U  e.  CHilOLD  /\  ( 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 24437 . 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, 4nvdir 24156 . 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 471 1  |-  ( ( U  e.  CHilOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
)  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   CCcc 9384    + caddc 9389   NrmCVeccnv 24107   +vcpv 24108   BaseSetcba 24109   .sOLDcns 24110   CHilOLDchlo 24431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-1st 6680  df-2nd 6681  df-vc 24069  df-nv 24115  df-va 24118  df-ba 24119  df-sm 24120  df-0v 24121  df-nmcv 24123  df-cbn 24409  df-hlo 24432
This theorem is referenced by:  axhvdistr2-zf  24538
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