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Theorem islindf 20427
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 6204 . . . . . 6 (𝑓 = 𝐹 → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝑓⟶(Base‘𝑤)))
21adantr 472 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝑓⟶(Base‘𝑤)))
3 dmeq 5492 . . . . . . 7 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
43adantr 472 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝑊) → dom 𝑓 = dom 𝐹)
5 fveq2 6375 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
6 islindf.b . . . . . . . 8 𝐵 = (Base‘𝑊)
75, 6syl6eqr 2817 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
87adantl 473 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝑊) → (Base‘𝑤) = 𝐵)
94, 8feq23d 6218 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝐹:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝐹𝐵))
102, 9bitrd 270 . . . 4 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝐹𝐵))
11 fvex 6388 . . . . . 6 (Scalar‘𝑤) ∈ V
12 fveq2 6375 . . . . . . . . 9 (𝑠 = (Scalar‘𝑤) → (Base‘𝑠) = (Base‘(Scalar‘𝑤)))
13 fveq2 6375 . . . . . . . . . 10 (𝑠 = (Scalar‘𝑤) → (0g𝑠) = (0g‘(Scalar‘𝑤)))
1413sneqd 4346 . . . . . . . . 9 (𝑠 = (Scalar‘𝑤) → {(0g𝑠)} = {(0g‘(Scalar‘𝑤))})
1512, 14difeq12d 3891 . . . . . . . 8 (𝑠 = (Scalar‘𝑤) → ((Base‘𝑠) ∖ {(0g𝑠)}) = ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}))
1615raleqdv 3292 . . . . . . 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 3133 . . . . . 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 3631 . . . . 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 6375 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
20 islindf.s . . . . . . . . . . . 12 𝑆 = (Scalar‘𝑊)
2119, 20syl6eqr 2817 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑆)
2221fveq2d 6379 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑆))
23 islindf.n . . . . . . . . . 10 𝑁 = (Base‘𝑆)
2422, 23syl6eqr 2817 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝑁)
2521fveq2d 6379 . . . . . . . . . . 11 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g𝑆))
26 islindf.z . . . . . . . . . . 11 0 = (0g𝑆)
2725, 26syl6eqr 2817 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
2827sneqd 4346 . . . . . . . . 9 (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
2924, 28difeq12d 3891 . . . . . . . 8 (𝑤 = 𝑊 → ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) = (𝑁 ∖ { 0 }))
3029adantl 473 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝑊) → ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}) = (𝑁 ∖ { 0 }))
31 fveq2 6375 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
32 islindf.v . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
3331, 32syl6eqr 2817 . . . . . . . . . . 11 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
3433adantl 473 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → ( ·𝑠𝑤) = · )
35 eqidd 2766 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → 𝑘 = 𝑘)
36 fveq1 6374 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
3736adantr 472 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓𝑥) = (𝐹𝑥))
3834, 35, 37oveq123d 6863 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑘( ·𝑠𝑤)(𝑓𝑥)) = (𝑘 · (𝐹𝑥)))
39 fveq2 6375 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
40 islindf.k . . . . . . . . . . . 12 𝐾 = (LSpan‘𝑊)
4139, 40syl6eqr 2817 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝐾)
4241adantl 473 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → (LSpan‘𝑤) = 𝐾)
43 imaeq1 5643 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝑓 ∖ {𝑥})))
443difeq1d 3889 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (dom 𝑓 ∖ {𝑥}) = (dom 𝐹 ∖ {𝑥}))
4544imaeq2d 5648 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝐹 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
4643, 45eqtrd 2799 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
4746adantr 472 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
4842, 47fveq12d 6382 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝑊) → ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) = (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
4938, 48eleq12d 2838 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝑊) → ((𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5049notbid 309 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝑊) → (¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5130, 50raleqbidv 3300 . . . . . 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 3300 . . . . 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, 52syl5bb 274 . . . 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 624 . . 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 20421 . . 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 5150 . 2 ((𝐹𝑋𝑊𝑌) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
5756ancoms 450 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  [wsbc 3596  cdif 3729  {csn 4334   class class class wbr 4809  dom cdm 5277  cima 5280  wf 6064  cfv 6068  (class class class)co 6842  Basecbs 16130  Scalarcsca 16217   ·𝑠 cvsca 16218  0gc0g 16366  LSpanclspn 19243   LIndF clindf 20419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-lindf 20421
This theorem is referenced by:  islinds2  20428  islindf2  20429  lindff  20430  lindfind  20431  f1lindf  20437  lsslindf  20445  matunitlindf  33831
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