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Theorem lindff 19973
Description: Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
lindff.b 𝐵 = (Base‘𝑊)
Assertion
Ref Expression
lindff ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹:dom 𝐹𝐵)

Proof of Theorem lindff
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . 3 ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹 LIndF 𝑊)
2 rellindf 19966 . . . . . 6 Rel LIndF
32brrelexi 5082 . . . . 5 (𝐹 LIndF 𝑊𝐹 ∈ V)
4 lindff.b . . . . . 6 𝐵 = (Base‘𝑊)
5 eqid 2610 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 eqid 2610 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
7 eqid 2610 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2610 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
9 eqid 2610 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
104, 5, 6, 7, 8, 9islindf 19970 . . . . 5 ((𝑊𝑌𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
113, 10sylan2 490 . . . 4 ((𝑊𝑌𝐹 LIndF 𝑊) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
1211ancoms 468 . . 3 ((𝐹 LIndF 𝑊𝑊𝑌) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
131, 12mpbid 221 . 2 ((𝐹 LIndF 𝑊𝑊𝑌) → (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
1413simpld 474 1 ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹:dom 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cdif 3537  {csn 4125   class class class wbr 4583  dom cdm 5038  cima 5041  wf 5800  cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  LSpanclspn 18792   LIndF clindf 19962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-lindf 19964
This theorem is referenced by:  lindfind2  19976  lindff1  19978  lindfrn  19979  f1lindf  19980  indlcim  19998
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