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Theorem nvdir 22114
 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
nvdir

Proof of Theorem nvdir
StepHypRef Expression
1 eqid 2438 . . 3
21nvvc 22096 . 2
3 nvdi.2 . . . 4
43vafval 22084 . . 3
5 nvdi.4 . . . 4
65smfval 22086 . . 3
7 nvdi.1 . . . 4
87, 3bafval 22085 . . 3
94, 6, 8vcdir 22034 . 2
102, 9sylan 459 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  cfv 5456  (class class class)co 6083  c1st 6349  cc 8990   caddc 8995  cvc 22026  cnv 22065  cpv 22066  cba 22067  cns 22068 This theorem is referenced by:  nvge0  22165  smcnlem  22195  ipidsq  22211  ip2i  22331  ipasslem1  22334  ipasslem11  22343  hldir  22412 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-1st 6351  df-2nd 6352  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-nmcv 22081
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