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Theorem lindff 21237
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 484 . . 3 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ π‘Œ) β†’ 𝐹 LIndF π‘Š)
2 rellindf 21230 . . . . . 6 Rel LIndF
32brrelex1i 5689 . . . . 5 (𝐹 LIndF π‘Š β†’ 𝐹 ∈ V)
4 lindff.b . . . . . 6 𝐡 = (Baseβ€˜π‘Š)
5 eqid 2733 . . . . . 6 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
6 eqid 2733 . . . . . 6 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
7 eqid 2733 . . . . . 6 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
8 eqid 2733 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
9 eqid 2733 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
104, 5, 6, 7, 8, 9islindf 21234 . . . . 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 594 . . . 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 460 . . 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 496 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   βˆ– cdif 3908  {csn 4587   class class class wbr 5106  dom cdm 5634   β€œ cima 5637  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326  LSpanclspn 20447   LIndF clindf 21226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-lindf 21228
This theorem is referenced by:  lindfind2  21240  lindff1  21242  lindfrn  21243  f1lindf  21244  indlcim  21262
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