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Theorem lindff 20809
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 486 . . 3 ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹 LIndF 𝑊)
2 rellindf 20802 . . . . . 6 Rel LIndF
32brrelex1i 5622 . . . . 5 (𝐹 LIndF 𝑊𝐹 ∈ V)
4 lindff.b . . . . . 6 𝐵 = (Base‘𝑊)
5 eqid 2739 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 eqid 2739 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
7 eqid 2739 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2739 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
9 eqid 2739 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
104, 5, 6, 7, 8, 9islindf 20806 . . . . 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 596 . . . 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 462 . . 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 235 . 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 498 1 ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹:dom 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  wral 3064  Vcvv 3423  cdif 3880  {csn 4557   class class class wbr 5069  dom cdm 5568  cima 5571  wf 6396  cfv 6400  (class class class)co 7234  Basecbs 16792  Scalarcsca 16837   ·𝑠 cvsca 16838  0gc0g 16976  LSpanclspn 20040   LIndF clindf 20798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-fv 6408  df-ov 7237  df-lindf 20800
This theorem is referenced by:  lindfind2  20812  lindff1  20814  lindfrn  20815  f1lindf  20816  indlcim  20834
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