MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lindff Structured version   Visualization version   GIF version
Theorem lindff 21591
Description: Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
lindff.b 𝐡 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
lindff ((𝐹 LIndF π‘Š ∧ π‘Š ∈ π‘Œ) β†’ 𝐹:dom 𝐹⟢𝐡)
Proof of Theorem lindff
Dummy variables π‘˜ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 481 . . 3 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ π‘Œ) β†’ 𝐹 LIndF π‘Š)
2 rellindf 21584 . . . . . 6 Rel LIndF
32brrelex1i 5733 . . . . 5 (𝐹 LIndF π‘Š β†’ 𝐹 ∈ V)
4 lindff.b . . . . . 6 𝐡 = (Baseβ€˜π‘Š)
5 eqid 2730 . . . . . 6 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
6 eqid 2730 . . . . . 6 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
7 eqid 2730 . . . . . 6 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
8 eqid 2730 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
9 eqid 2730 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
104, 5, 6, 7, 8, 9islindf 21588 . . . . 5 ((π‘Š ∈ π‘Œ ∧ 𝐹 ∈ V) β†’ (𝐹 LIndF π‘Š ↔ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
113, 10sylan2 591 . . . 4 ((π‘Š ∈ π‘Œ ∧ 𝐹 LIndF π‘Š) β†’ (𝐹 LIndF π‘Š ↔ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
1211ancoms 457 . . 3 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ π‘Œ) β†’ (𝐹 LIndF π‘Š ↔ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
131, 12mpbid 231 . 2 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ π‘Œ) β†’ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
1413simpld 493 1 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ π‘Œ) β†’ 𝐹:dom 𝐹⟢𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472   βˆ– cdif 3946  {csn 4629   class class class wbr 5149  dom cdm 5677   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413  Basecbs 17150  Scalarcsca 17206   ·𝑠 cvsca 17207  0gc0g 17391  LSpanclspn 20728   LIndF clindf 21580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7416  df-lindf 21582
This theorem is referenced by:  lindfind2  21594  lindff1  21596  lindfrn  21597  f1lindf  21598  indlcim  21616
  Copyright terms: Public domain W3C validator