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Theorem nvdi 22116
 Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvdi.1
nvdi.2
nvdi.4
Assertion
Ref Expression
nvdi

Proof of Theorem nvdi
StepHypRef Expression
1 eqid 2438 . . 3
21nvvc 22099 . 2
3 nvdi.2 . . . 4
43vafval 22087 . . 3
5 nvdi.4 . . . 4
65smfval 22089 . . 3
7 nvdi.1 . . . 4
87, 3bafval 22088 . . 3
94, 6, 8vcdi 22036 . 2
102, 9sylan 459 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  cfv 5457  (class class class)co 6084  c1st 6350  cc 8993  cvc 22029  cnv 22068  cpv 22069  cba 22070  cns 22071 This theorem is referenced by:  nvmdi  22136  nvaddsub4  22147  nvnncan  22149  nvdif  22159  nvpi  22160  ipdirilem  22335  hldi  22414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-1st 6352  df-2nd 6353  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-nmcv 22084
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