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Theorem lindff 21361
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 483 . . 3 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ π‘Œ) β†’ 𝐹 LIndF π‘Š)
2 rellindf 21354 . . . . . 6 Rel LIndF
32brrelex1i 5730 . . . . 5 (𝐹 LIndF π‘Š β†’ 𝐹 ∈ V)
4 lindff.b . . . . . 6 𝐡 = (Baseβ€˜π‘Š)
5 eqid 2732 . . . . . 6 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
6 eqid 2732 . . . . . 6 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
7 eqid 2732 . . . . . 6 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
8 eqid 2732 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
9 eqid 2732 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
104, 5, 6, 7, 8, 9islindf 21358 . . . . 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 593 . . . 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 459 . . 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 495 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆ– cdif 3944  {csn 4627   class class class wbr 5147  dom cdm 5675   β€œ cima 5678  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  LSpanclspn 20574   LIndF clindf 21350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-lindf 21352
This theorem is referenced by:  lindfind2  21364  lindff1  21366  lindfrn  21367  f1lindf  21368  indlcim  21386
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