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Theorem lindff 21853
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 482 . . 3 ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹 LIndF 𝑊)
2 rellindf 21846 . . . . . 6 Rel LIndF
32brrelex1i 5745 . . . . 5 (𝐹 LIndF 𝑊𝐹 ∈ V)
4 lindff.b . . . . . 6 𝐵 = (Base‘𝑊)
5 eqid 2735 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 eqid 2735 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
7 eqid 2735 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2735 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
9 eqid 2735 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
104, 5, 6, 7, 8, 9islindf 21850 . . . . 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 593 . . . 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 458 . . 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 232 . 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 494 1 ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹:dom 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cdif 3960  {csn 4631   class class class wbr 5148  dom cdm 5689  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486  LSpanclspn 20987   LIndF clindf 21842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-lindf 21844
This theorem is referenced by:  lindfind2  21856  lindff1  21858  lindfrn  21859  f1lindf  21860  indlcim  21878
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