MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lindff Structured version   Visualization version   GIF version

Theorem lindff 20094
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 473 . . 3 ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹 LIndF 𝑊)
2 rellindf 20087 . . . . . 6 Rel LIndF
32brrelexi 5128 . . . . 5 (𝐹 LIndF 𝑊𝐹 ∈ V)
4 lindff.b . . . . . 6 𝐵 = (Base‘𝑊)
5 eqid 2621 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 eqid 2621 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
7 eqid 2621 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2621 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
9 eqid 2621 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
104, 5, 6, 7, 8, 9islindf 20091 . . . . 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 491 . . . 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 469 . . 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 222 . 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 475 1 ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹:dom 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  cdif 3557  {csn 4155   class class class wbr 4623  dom cdm 5084  cima 5087  wf 5853  cfv 5857  (class class class)co 6615  Basecbs 15800  Scalarcsca 15884   ·𝑠 cvsca 15885  0gc0g 16040  LSpanclspn 18911   LIndF clindf 20083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-lindf 20085
This theorem is referenced by:  lindfind2  20097  lindff1  20099  lindfrn  20100  f1lindf  20101  indlcim  20119
  Copyright terms: Public domain W3C validator