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Theorem lindff 27253
 Description: Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
lindff.b
Assertion
Ref Expression
lindff LIndF

Proof of Theorem lindff
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . 3 LIndF LIndF
2 rellindf 27246 . . . . . 6 LIndF
32brrelexi 4910 . . . . 5 LIndF
4 lindff.b . . . . . 6
5 eqid 2435 . . . . . 6
6 eqid 2435 . . . . . 6
7 eqid 2435 . . . . . 6 Scalar Scalar
8 eqid 2435 . . . . . 6 Scalar Scalar
9 eqid 2435 . . . . . 6 Scalar Scalar
104, 5, 6, 7, 8, 9islindf 27250 . . . . 5 LIndF Scalar Scalar
113, 10sylan2 461 . . . 4 LIndF LIndF Scalar Scalar
1211ancoms 440 . . 3 LIndF LIndF Scalar Scalar
131, 12mpbid 202 . 2 LIndF Scalar Scalar
1413simpld 446 1 LIndF
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  cvv 2948   cdif 3309  csn 3806   class class class wbr 4204   cdm 4870  cima 4873  wf 5442  cfv 5446  (class class class)co 6073  cbs 13461  Scalarcsca 13524  cvsca 13525  c0g 13715  clspn 16039   LIndF clindf 27242 This theorem is referenced by:  lindfind2  27256  lindff1  27258  lindfrn  27259  f1lindf  27260  indlcim  27278 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-lindf 27244
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