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Theorem islindf 21234
Description: Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
islindf.b 𝐡 = (Baseβ€˜π‘Š)
islindf.v Β· = ( ·𝑠 β€˜π‘Š)
islindf.k 𝐾 = (LSpanβ€˜π‘Š)
islindf.s 𝑆 = (Scalarβ€˜π‘Š)
islindf.n 𝑁 = (Baseβ€˜π‘†)
islindf.z 0 = (0gβ€˜π‘†)
Assertion
Ref Expression
islindf ((π‘Š ∈ π‘Œ ∧ 𝐹 ∈ 𝑋) β†’ (𝐹 LIndF π‘Š ↔ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ (𝑁 βˆ– { 0 }) Β¬ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
Distinct variable groups:   π‘˜,𝐹,π‘₯   π‘˜,𝑁   π‘˜,π‘Š,π‘₯   0 ,π‘˜
Allowed substitution hints:   𝐡(π‘₯,π‘˜)   𝑆(π‘₯,π‘˜)   Β· (π‘₯,π‘˜)   𝐾(π‘₯,π‘˜)   𝑁(π‘₯)   𝑋(π‘₯,π‘˜)   π‘Œ(π‘₯,π‘˜)   0 (π‘₯)

Proof of Theorem islindf
Dummy variables 𝑓 𝑀 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 6650 . . . . . 6 (𝑓 = 𝐹 β†’ (𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ↔ 𝐹:dom π‘“βŸΆ(Baseβ€˜π‘€)))
21adantr 482 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ↔ 𝐹:dom π‘“βŸΆ(Baseβ€˜π‘€)))
3 dmeq 5860 . . . . . . 7 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
43adantr 482 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ dom 𝑓 = dom 𝐹)
5 fveq2 6843 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
6 islindf.b . . . . . . . 8 𝐡 = (Baseβ€˜π‘Š)
75, 6eqtr4di 2791 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝐡)
87adantl 483 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (Baseβ€˜π‘€) = 𝐡)
94, 8feq23d 6664 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (𝐹:dom π‘“βŸΆ(Baseβ€˜π‘€) ↔ 𝐹:dom 𝐹⟢𝐡))
102, 9bitrd 279 . . . 4 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ↔ 𝐹:dom 𝐹⟢𝐡))
11 fvex 6856 . . . . . 6 (Scalarβ€˜π‘€) ∈ V
12 fveq2 6843 . . . . . . . . 9 (𝑠 = (Scalarβ€˜π‘€) β†’ (Baseβ€˜π‘ ) = (Baseβ€˜(Scalarβ€˜π‘€)))
13 fveq2 6843 . . . . . . . . . 10 (𝑠 = (Scalarβ€˜π‘€) β†’ (0gβ€˜π‘ ) = (0gβ€˜(Scalarβ€˜π‘€)))
1413sneqd 4599 . . . . . . . . 9 (𝑠 = (Scalarβ€˜π‘€) β†’ {(0gβ€˜π‘ )} = {(0gβ€˜(Scalarβ€˜π‘€))})
1512, 14difeq12d 4084 . . . . . . . 8 (𝑠 = (Scalarβ€˜π‘€) β†’ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) = ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}))
1615raleqdv 3312 . . . . . . 7 (𝑠 = (Scalarβ€˜π‘€) β†’ (βˆ€π‘˜ ∈ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) ↔ βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})))))
1716ralbidv 3171 . . . . . 6 (𝑠 = (Scalarβ€˜π‘€) β†’ (βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) ↔ βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})))))
1811, 17sbcie 3783 . . . . 5 ([(Scalarβ€˜π‘€) / 𝑠]βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) ↔ βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))))
19 fveq2 6843 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
20 islindf.s . . . . . . . . . . . 12 𝑆 = (Scalarβ€˜π‘Š)
2119, 20eqtr4di 2791 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝑆)
2221fveq2d 6847 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜π‘†))
23 islindf.n . . . . . . . . . 10 𝑁 = (Baseβ€˜π‘†)
2422, 23eqtr4di 2791 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = 𝑁)
2521fveq2d 6847 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜π‘†))
26 islindf.z . . . . . . . . . . 11 0 = (0gβ€˜π‘†)
2725, 26eqtr4di 2791 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (0gβ€˜(Scalarβ€˜π‘€)) = 0 )
2827sneqd 4599 . . . . . . . . 9 (𝑀 = π‘Š β†’ {(0gβ€˜(Scalarβ€˜π‘€))} = { 0 })
2924, 28difeq12d 4084 . . . . . . . 8 (𝑀 = π‘Š β†’ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) = (𝑁 βˆ– { 0 }))
3029adantl 483 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) = (𝑁 βˆ– { 0 }))
31 fveq2 6843 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘Š))
32 islindf.v . . . . . . . . . . . 12 Β· = ( ·𝑠 β€˜π‘Š)
3331, 32eqtr4di 2791 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = Β· )
3433adantl 483 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ ( ·𝑠 β€˜π‘€) = Β· )
35 eqidd 2734 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ π‘˜ = π‘˜)
36 fveq1 6842 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
3736adantr 482 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
3834, 35, 37oveq123d 7379 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) = (π‘˜ Β· (πΉβ€˜π‘₯)))
39 fveq2 6843 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = (LSpanβ€˜π‘Š))
40 islindf.k . . . . . . . . . . . 12 𝐾 = (LSpanβ€˜π‘Š)
4139, 40eqtr4di 2791 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = 𝐾)
4241adantl 483 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (LSpanβ€˜π‘€) = 𝐾)
43 imaeq1 6009 . . . . . . . . . . . 12 (𝑓 = 𝐹 β†’ (𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})) = (𝐹 β€œ (dom 𝑓 βˆ– {π‘₯})))
443difeq1d 4082 . . . . . . . . . . . . 13 (𝑓 = 𝐹 β†’ (dom 𝑓 βˆ– {π‘₯}) = (dom 𝐹 βˆ– {π‘₯}))
4544imaeq2d 6014 . . . . . . . . . . . 12 (𝑓 = 𝐹 β†’ (𝐹 β€œ (dom 𝑓 βˆ– {π‘₯})) = (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))
4643, 45eqtrd 2773 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ (𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})) = (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))
4746adantr 482 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})) = (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))
4842, 47fveq12d 6850 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) = (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))
4938, 48eleq12d 2828 . . . . . . . 8 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ ((π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) ↔ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
5049notbid 318 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) ↔ Β¬ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
5130, 50raleqbidv 3318 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) ↔ βˆ€π‘˜ ∈ (𝑁 βˆ– { 0 }) Β¬ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
524, 51raleqbidv 3318 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) ↔ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ (𝑁 βˆ– { 0 }) Β¬ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
5318, 52bitrid 283 . . . 4 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ ([(Scalarβ€˜π‘€) / 𝑠]βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))) ↔ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ (𝑁 βˆ– { 0 }) Β¬ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
5410, 53anbi12d 632 . . 3 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ ((𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ∧ [(Scalarβ€˜π‘€) / 𝑠]βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})))) ↔ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ (𝑁 βˆ– { 0 }) Β¬ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
55 df-lindf 21228 . . 3 LIndF = {βŸ¨π‘“, π‘€βŸ© ∣ (𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ∧ [(Scalarβ€˜π‘€) / 𝑠]βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))))}
5654, 55brabga 5492 . 2 ((𝐹 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐹 LIndF π‘Š ↔ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ (𝑁 βˆ– { 0 }) Β¬ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
5756ancoms 460 1 ((π‘Š ∈ π‘Œ ∧ 𝐹 ∈ 𝑋) β†’ (𝐹 LIndF π‘Š ↔ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ (𝑁 βˆ– { 0 }) Β¬ (π‘˜ Β· (πΉβ€˜π‘₯)) ∈ (πΎβ€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
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  [wsbc 3740   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-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:  islinds2  21235  islindf2  21236  lindff  21237  lindfind  21238  f1lindf  21244  lsslindf  21252  lindfpropd  32217  matunitlindf  36122
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