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Theorem linds0 32548
 Description: The empty set is always a linearly independet subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
linds0 linIndS

Proof of Theorem linds0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3938 . . . . . 6 Scalar
21a1ii 27 . . . . 5 finSupp Scalar linC Scalar
3 0ex 4583 . . . . . 6
4 breq1 4456 . . . . . . . . 9 finSupp Scalar finSupp Scalar
5 oveq1 6302 . . . . . . . . . 10 linC linC
65eqeq1d 2469 . . . . . . . . 9 linC linC
74, 6anbi12d 710 . . . . . . . 8 finSupp Scalar linC finSupp Scalar linC
8 fveq1 5871 . . . . . . . . . 10
98eqeq1d 2469 . . . . . . . . 9 Scalar Scalar
109ralbidv 2906 . . . . . . . 8 Scalar Scalar
117, 10imbi12d 320 . . . . . . 7 finSupp Scalar linC Scalar finSupp Scalar linC Scalar
1211ralsng 4068 . . . . . 6 finSupp Scalar linC Scalar finSupp Scalar linC Scalar
133, 12mp1i 12 . . . . 5 finSupp Scalar linC Scalar finSupp Scalar linC Scalar
142, 13mpbird 232 . . . 4 finSupp Scalar linC Scalar
15 fvex 5882 . . . . . . 7 Scalar
16 map0e 7468 . . . . . . 7 Scalar Scalar
1715, 16mp1i 12 . . . . . 6 Scalar
18 df1o2 7154 . . . . . 6
1917, 18syl6eq 2524 . . . . 5 Scalar
2019raleqdv 3069 . . . 4 Scalar finSupp Scalar linC Scalar finSupp Scalar linC Scalar
2114, 20mpbird 232 . . 3 Scalar finSupp Scalar linC Scalar
22 0elpw 4622 . . 3
2321, 22jctil 537 . 2 Scalar finSupp Scalar linC Scalar
24 eqid 2467 . . . 4
25 eqid 2467 . . . 4
26 eqid 2467 . . . 4 Scalar Scalar
27 eqid 2467 . . . 4 Scalar Scalar
28 eqid 2467 . . . 4 Scalar Scalar
2924, 25, 26, 27, 28islininds 32529 . . 3 linIndS Scalar finSupp Scalar linC Scalar
303, 29mpan 670 . 2 linIndS Scalar finSupp Scalar linC Scalar
3123, 30mpbird 232 1 linIndS
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  wral 2817  cvv 3118  c0 3790  cpw 4016  csn 4033   class class class wbr 4453  cfv 5594  (class class class)co 6295  c1o 7135   cmap 7432   finSupp cfsupp 7841  cbs 14507  Scalarcsca 14575  c0g 14712   linC clinc 32487   linIndS clininds 32523 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-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-suc 4890  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-oprab 6299  df-mpt2 6300  df-1o 7142  df-map 7434  df-lininds 32525 This theorem is referenced by: (None)
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