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Theorem islindf 21366
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 6698 . . . . . 6 (𝑓 = 𝐹 β†’ (𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ↔ 𝐹:dom π‘“βŸΆ(Baseβ€˜π‘€)))
21adantr 481 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ↔ 𝐹:dom π‘“βŸΆ(Baseβ€˜π‘€)))
3 dmeq 5903 . . . . . . 7 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
43adantr 481 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ dom 𝑓 = dom 𝐹)
5 fveq2 6891 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
6 islindf.b . . . . . . . 8 𝐡 = (Baseβ€˜π‘Š)
75, 6eqtr4di 2790 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝐡)
87adantl 482 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (Baseβ€˜π‘€) = 𝐡)
94, 8feq23d 6712 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (𝐹:dom π‘“βŸΆ(Baseβ€˜π‘€) ↔ 𝐹:dom 𝐹⟢𝐡))
102, 9bitrd 278 . . . 4 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ↔ 𝐹:dom 𝐹⟢𝐡))
11 fvex 6904 . . . . . 6 (Scalarβ€˜π‘€) ∈ V
12 fveq2 6891 . . . . . . . . 9 (𝑠 = (Scalarβ€˜π‘€) β†’ (Baseβ€˜π‘ ) = (Baseβ€˜(Scalarβ€˜π‘€)))
13 fveq2 6891 . . . . . . . . . 10 (𝑠 = (Scalarβ€˜π‘€) β†’ (0gβ€˜π‘ ) = (0gβ€˜(Scalarβ€˜π‘€)))
1413sneqd 4640 . . . . . . . . 9 (𝑠 = (Scalarβ€˜π‘€) β†’ {(0gβ€˜π‘ )} = {(0gβ€˜(Scalarβ€˜π‘€))})
1512, 14difeq12d 4123 . . . . . . . 8 (𝑠 = (Scalarβ€˜π‘€) β†’ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) = ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}))
1615raleqdv 3325 . . . . . . 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 3177 . . . . . 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 3820 . . . . 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 6891 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
20 islindf.s . . . . . . . . . . . 12 𝑆 = (Scalarβ€˜π‘Š)
2119, 20eqtr4di 2790 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝑆)
2221fveq2d 6895 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜π‘†))
23 islindf.n . . . . . . . . . 10 𝑁 = (Baseβ€˜π‘†)
2422, 23eqtr4di 2790 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = 𝑁)
2521fveq2d 6895 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜π‘†))
26 islindf.z . . . . . . . . . . 11 0 = (0gβ€˜π‘†)
2725, 26eqtr4di 2790 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (0gβ€˜(Scalarβ€˜π‘€)) = 0 )
2827sneqd 4640 . . . . . . . . 9 (𝑀 = π‘Š β†’ {(0gβ€˜(Scalarβ€˜π‘€))} = { 0 })
2924, 28difeq12d 4123 . . . . . . . 8 (𝑀 = π‘Š β†’ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) = (𝑁 βˆ– { 0 }))
3029adantl 482 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ ((Baseβ€˜(Scalarβ€˜π‘€)) βˆ– {(0gβ€˜(Scalarβ€˜π‘€))}) = (𝑁 βˆ– { 0 }))
31 fveq2 6891 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘Š))
32 islindf.v . . . . . . . . . . . 12 Β· = ( ·𝑠 β€˜π‘Š)
3331, 32eqtr4di 2790 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = Β· )
3433adantl 482 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ ( ·𝑠 β€˜π‘€) = Β· )
35 eqidd 2733 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ π‘˜ = π‘˜)
36 fveq1 6890 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
3736adantr 481 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
3834, 35, 37oveq123d 7429 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) = (π‘˜ Β· (πΉβ€˜π‘₯)))
39 fveq2 6891 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = (LSpanβ€˜π‘Š))
40 islindf.k . . . . . . . . . . . 12 𝐾 = (LSpanβ€˜π‘Š)
4139, 40eqtr4di 2790 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = 𝐾)
4241adantl 482 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (LSpanβ€˜π‘€) = 𝐾)
43 imaeq1 6054 . . . . . . . . . . . 12 (𝑓 = 𝐹 β†’ (𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})) = (𝐹 β€œ (dom 𝑓 βˆ– {π‘₯})))
443difeq1d 4121 . . . . . . . . . . . . 13 (𝑓 = 𝐹 β†’ (dom 𝑓 βˆ– {π‘₯}) = (dom 𝐹 βˆ– {π‘₯}))
4544imaeq2d 6059 . . . . . . . . . . . 12 (𝑓 = 𝐹 β†’ (𝐹 β€œ (dom 𝑓 βˆ– {π‘₯})) = (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))
4643, 45eqtrd 2772 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ (𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})) = (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))
4746adantr 481 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = π‘Š) β†’ (𝑓 β€œ (dom 𝑓 βˆ– {π‘₯})) = (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))
4842, 47fveq12d 6898 . . . . . . . . 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 2827 . . . . . . . 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 317 . . . . . . 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 3342 . . . . . 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 3342 . . . . 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 282 . . . 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 631 . . 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 21360 . . 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 5534 . 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 459 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  [wsbc 3777   βˆ– cdif 3945  {csn 4628   class class class wbr 5148  dom cdm 5676   β€œ cima 5679  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  Scalarcsca 17199   ·𝑠 cvsca 17200  0gc0g 17384  LSpanclspn 20581   LIndF clindf 21358
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 5299  ax-nul 5306  ax-pr 5427
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-lindf 21360
This theorem is referenced by:  islinds2  21367  islindf2  21368  lindff  21369  lindfind  21370  f1lindf  21376  lsslindf  21384  lindfpropd  32493  matunitlindf  36481
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