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Theorem vcdi 26016
 Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vci.1
vci.2
vci.3
Assertion
Ref Expression
vcdi

Proof of Theorem vcdi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vci.1 . . . . . 6
2 vci.2 . . . . . 6
3 vci.3 . . . . . 6
41, 2, 3vci 26012 . . . . 5
5 simpl 458 . . . . . . . . 9
65ralimi 2825 . . . . . . . 8
76adantl 467 . . . . . . 7
87ralimi 2825 . . . . . 6
983ad2ant3 1028 . . . . 5
104, 9syl 17 . . . 4
11 oveq1 6312 . . . . . . 7
1211oveq2d 6321 . . . . . 6
13 oveq2 6313 . . . . . . 7
1413oveq1d 6320 . . . . . 6
1512, 14eqeq12d 2451 . . . . 5
16 oveq1 6312 . . . . . 6
17 oveq1 6312 . . . . . . 7
18 oveq1 6312 . . . . . . 7
1917, 18oveq12d 6323 . . . . . 6
2016, 19eqeq12d 2451 . . . . 5
21 oveq2 6313 . . . . . . 7
2221oveq2d 6321 . . . . . 6
23 oveq2 6313 . . . . . . 7
2423oveq2d 6321 . . . . . 6
2522, 24eqeq12d 2451 . . . . 5
2615, 20, 25rspc3v 3200 . . . 4
2710, 26syl5 33 . . 3
28273com12 1209 . 2
2928impcom 431 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437   wcel 1870  wral 2782   cxp 4852   crn 4855  wf 5597  cfv 5601  (class class class)co 6305  c1st 6805  c2nd 6806  cc 9536  c1 9539   caddc 9541   cmul 9543  cablo 25854  cvc 26009 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-1st 6807  df-2nd 6808  df-vc 26010 This theorem is referenced by:  vcnegsubdi2  26039  vcsub4  26040  nvdi  26096
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