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Theorem islindf 21300
Description: Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
islindf.b 𝐵 = (Base‘𝑊)
islindf.v · = ( ·𝑠𝑊)
islindf.k 𝐾 = (LSpan‘𝑊)
islindf.s 𝑆 = (Scalar‘𝑊)
islindf.n 𝑁 = (Base‘𝑆)
islindf.z 0 = (0g𝑆)
Assertion
Ref Expression
islindf ((𝑊𝑌𝐹𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑘,𝑁   𝑘,𝑊,𝑥   0 ,𝑘
Allowed substitution hints:   𝐵(𝑥,𝑘)   𝑆(𝑥,𝑘)   · (𝑥,𝑘)   𝐾(𝑥,𝑘)   𝑁(𝑥)   𝑋(𝑥,𝑘)   𝑌(𝑥,𝑘)   0 (𝑥)

Proof of Theorem islindf
Dummy variables 𝑓 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 6685 . . . . . 6 (𝑓 = 𝐹 → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝑓⟶(Base‘𝑤)))
21adantr 481 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝑓⟶(Base‘𝑤)))
3 dmeq 5895 . . . . . . 7 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
43adantr 481 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝑊) → dom 𝑓 = dom 𝐹)
5 fveq2 6878 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
6 islindf.b . . . . . . . 8 𝐵 = (Base‘𝑊)
75, 6eqtr4di 2789 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
87adantl 482 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝑊) → (Base‘𝑤) = 𝐵)
94, 8feq23d 6699 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝐹:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝐹𝐵))
102, 9bitrd 278 . . . 4 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝐹𝐵))
11 fvex 6891 . . . . . 6 (Scalar‘𝑤) ∈ V
12 fveq2 6878 . . . . . . . . 9 (𝑠 = (Scalar‘𝑤) → (Base‘𝑠) = (Base‘(Scalar‘𝑤)))
13 fveq2 6878 . . . . . . . . . 10 (𝑠 = (Scalar‘𝑤) → (0g𝑠) = (0g‘(Scalar‘𝑤)))
1413sneqd 4634 . . . . . . . . 9 (𝑠 = (Scalar‘𝑤) → {(0g𝑠)} = {(0g‘(Scalar‘𝑤))})
1512, 14difeq12d 4119 . . . . . . . 8 (𝑠 = (Scalar‘𝑤) → ((Base‘𝑠) ∖ {(0g𝑠)}) = ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}))
1615raleqdv 3324 . . . . . . 7 (𝑠 = (Scalar‘𝑤) → (∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))))
1716ralbidv 3176 . . . . . 6 (𝑠 = (Scalar‘𝑤) → (∀𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))))
1811, 17sbcie 3816 . . . . 5 ([(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))
19 fveq2 6878 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
20 islindf.s . . . . . . . . . . . 12 𝑆 = (Scalar‘𝑊)
2119, 20eqtr4di 2789 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑆)
2221fveq2d 6882 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑆))
23 islindf.n . . . . . . . . . 10 𝑁 = (Base‘𝑆)
2422, 23eqtr4di 2789 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝑁)
2521fveq2d 6882 . . . . . . . . . . 11 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g𝑆))
26 islindf.z . . . . . . . . . . 11 0 = (0g𝑆)
2725, 26eqtr4di 2789 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
2827sneqd 4634 . . . . . . . . 9 (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
2924, 28difeq12d 4119 . . . . . . . 8 (𝑤 = 𝑊 → ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) = (𝑁 ∖ { 0 }))
3029adantl 482 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝑊) → ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) = (𝑁 ∖ { 0 }))
31 fveq2 6878 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
32 islindf.v . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
3331, 32eqtr4di 2789 . . . . . . . . . . 11 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
3433adantl 482 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → ( ·𝑠𝑤) = · )
35 eqidd 2732 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → 𝑘 = 𝑘)
36 fveq1 6877 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
3736adantr 481 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓𝑥) = (𝐹𝑥))
3834, 35, 37oveq123d 7414 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑘( ·𝑠𝑤)(𝑓𝑥)) = (𝑘 · (𝐹𝑥)))
39 fveq2 6878 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
40 islindf.k . . . . . . . . . . . 12 𝐾 = (LSpan‘𝑊)
4139, 40eqtr4di 2789 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝐾)
4241adantl 482 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → (LSpan‘𝑤) = 𝐾)
43 imaeq1 6044 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝑓 ∖ {𝑥})))
443difeq1d 4117 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (dom 𝑓 ∖ {𝑥}) = (dom 𝐹 ∖ {𝑥}))
4544imaeq2d 6049 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝐹 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
4643, 45eqtrd 2771 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
4746adantr 481 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
4842, 47fveq12d 6885 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝑊) → ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) = (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
4938, 48eleq12d 2826 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝑊) → ((𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5049notbid 317 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝑊) → (¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5130, 50raleqbidv 3341 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝑊) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
524, 51raleqbidv 3341 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝑊) → (∀𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5318, 52bitrid 282 . . . 4 ((𝑓 = 𝐹𝑤 = 𝑊) → ([(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5410, 53anbi12d 631 . . 3 ((𝑓 = 𝐹𝑤 = 𝑊) → ((𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))) ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
55 df-lindf 21294 . . 3 LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
5654, 55brabga 5527 . 2 ((𝐹𝑋𝑊𝑌) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
5756ancoms 459 1 ((𝑊𝑌𝐹𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  [wsbc 3773  cdif 3941  {csn 4622   class class class wbr 5141  dom cdm 5669  cima 5672  wf 6528  cfv 6532  (class class class)co 7393  Basecbs 17126  Scalarcsca 17182   ·𝑠 cvsca 17183  0gc0g 17367  LSpanclspn 20531   LIndF clindf 21292
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-lindf 21294
This theorem is referenced by:  islinds2  21301  islindf2  21302  lindff  21303  lindfind  21304  f1lindf  21310  lsslindf  21318  lindfpropd  32356  matunitlindf  36288
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