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Theorem vci 25272
 Description: The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable was chosen because is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vci.1
vci.2
vci.3
Assertion
Ref Expression
vci
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem vci
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vci.1 . . . . 5
21eqeq2i 2485 . . . 4
3 eleq1 2539 . . . . 5
4 rneq 5234 . . . . . . 7
5 vci.3 . . . . . . 7
64, 5syl6eqr 2526 . . . . . 6
7 xpeq2 5020 . . . . . . . 8
87feq2d 5724 . . . . . . 7
9 feq3 5721 . . . . . . 7
108, 9bitrd 253 . . . . . 6
116, 10syl 16 . . . . 5
12 oveq 6301 . . . . . . . . . . . 12
1312oveq2d 6311 . . . . . . . . . . 11
14 oveq 6301 . . . . . . . . . . 11
1513, 14eqeq12d 2489 . . . . . . . . . 10
166, 15raleqbidv 3077 . . . . . . . . 9
17 oveq 6301 . . . . . . . . . . . 12
1817eqeq2d 2481 . . . . . . . . . . 11
1918anbi1d 704 . . . . . . . . . 10
2019ralbidv 2906 . . . . . . . . 9
2116, 20anbi12d 710 . . . . . . . 8
2221ralbidv 2906 . . . . . . 7
2322anbi2d 703 . . . . . 6
246, 23raleqbidv 3077 . . . . 5
253, 11, 243anbi123d 1299 . . . 4
262, 25sylbir 213 . . 3
27 vci.2 . . . . 5
2827eqeq2i 2485 . . . 4
29 feq1 5719 . . . . 5
30 oveq 6301 . . . . . . . 8
3130eqeq1d 2469 . . . . . . 7
32 oveq 6301 . . . . . . . . . . 11
33 oveq 6301 . . . . . . . . . . . 12
34 oveq 6301 . . . . . . . . . . . 12
3533, 34oveq12d 6313 . . . . . . . . . . 11
3632, 35eqeq12d 2489 . . . . . . . . . 10
3736ralbidv 2906 . . . . . . . . 9
38 oveq 6301 . . . . . . . . . . . 12
39 oveq 6301 . . . . . . . . . . . . 13
4033, 39oveq12d 6313 . . . . . . . . . . . 12
4138, 40eqeq12d 2489 . . . . . . . . . . 11
42 oveq 6301 . . . . . . . . . . . 12
4339oveq2d 6311 . . . . . . . . . . . . 13
44 oveq 6301 . . . . . . . . . . . . 13
4543, 44eqtrd 2508 . . . . . . . . . . . 12
4642, 45eqeq12d 2489 . . . . . . . . . . 11
4741, 46anbi12d 710 . . . . . . . . . 10
4847ralbidv 2906 . . . . . . . . 9
4937, 48anbi12d 710 . . . . . . . 8
5049ralbidv 2906 . . . . . . 7
5131, 50anbi12d 710 . . . . . 6
5251ralbidv 2906 . . . . 5
5329, 523anbi23d 1302 . . . 4
5428, 53sylbir 213 . . 3
5526, 54elopabi 6856 . 2
56 df-vc 25270 . 2
5755, 56eleq2s 2575 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1379   wcel 1767  wral 2817  copab 4510   cxp 5003   crn 5006  wf 5590  cfv 5594  (class class class)co 6295  c1st 6793  c2nd 6794  cc 9502  c1 9505   caddc 9507   cmul 9509  cablo 25114  cvc 25269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-1st 6795  df-2nd 6796  df-vc 25270 This theorem is referenced by:  vcsm  25273  vcid  25275  vcdi  25276  vcdir  25277  vcass  25278  vcablo  25281
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