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Theorem islindf 20353
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 6187 . . . . . 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 5479 . . . . . . 7 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
43adantr 472 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝑊) → dom 𝑓 = dom 𝐹)
5 fveq2 6352 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
6 islindf.b . . . . . . . 8 𝐵 = (Base‘𝑊)
75, 6syl6eqr 2812 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
87adantl 473 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝑊) → (Base‘𝑤) = 𝐵)
94, 8feq23d 6201 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝐹:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝐹𝐵))
102, 9bitrd 268 . . . 4 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓:dom 𝑓⟶(Base‘𝑤) ↔ 𝐹:dom 𝐹𝐵))
11 fvex 6362 . . . . . 6 (Scalar‘𝑤) ∈ V
12 fveq2 6352 . . . . . . . . 9 (𝑠 = (Scalar‘𝑤) → (Base‘𝑠) = (Base‘(Scalar‘𝑤)))
13 fveq2 6352 . . . . . . . . . 10 (𝑠 = (Scalar‘𝑤) → (0g𝑠) = (0g‘(Scalar‘𝑤)))
1413sneqd 4333 . . . . . . . . 9 (𝑠 = (Scalar‘𝑤) → {(0g𝑠)} = {(0g‘(Scalar‘𝑤))})
1512, 14difeq12d 3872 . . . . . . . 8 (𝑠 = (Scalar‘𝑤) → ((Base‘𝑠) ∖ {(0g𝑠)}) = ((Base‘(Scalar‘𝑤)) ∖ {(0g‘(Scalar‘𝑤))}))
1615raleqdv 3283 . . . . . . 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 3124 . . . . . 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 3611 . . . . 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 6352 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
20 islindf.s . . . . . . . . . . . 12 𝑆 = (Scalar‘𝑊)
2119, 20syl6eqr 2812 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑆)
2221fveq2d 6356 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑆))
23 islindf.n . . . . . . . . . 10 𝑁 = (Base‘𝑆)
2422, 23syl6eqr 2812 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝑁)
2521fveq2d 6356 . . . . . . . . . . 11 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g𝑆))
26 islindf.z . . . . . . . . . . 11 0 = (0g𝑆)
2725, 26syl6eqr 2812 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
2827sneqd 4333 . . . . . . . . 9 (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
2924, 28difeq12d 3872 . . . . . . . 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 6352 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
32 islindf.v . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
3331, 32syl6eqr 2812 . . . . . . . . . . 11 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
3433adantl 473 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → ( ·𝑠𝑤) = · )
35 eqidd 2761 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → 𝑘 = 𝑘)
36 fveq1 6351 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
3736adantr 472 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑓𝑥) = (𝐹𝑥))
3834, 35, 37oveq123d 6834 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝑊) → (𝑘( ·𝑠𝑤)(𝑓𝑥)) = (𝑘 · (𝐹𝑥)))
39 fveq2 6352 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
40 islindf.k . . . . . . . . . . . 12 𝐾 = (LSpan‘𝑊)
4139, 40syl6eqr 2812 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝐾)
4241adantl 473 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝑊) → (LSpan‘𝑤) = 𝐾)
43 imaeq1 5619 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝑓 ∖ {𝑥})))
443difeq1d 3870 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (dom 𝑓 ∖ {𝑥}) = (dom 𝐹 ∖ {𝑥}))
4544imaeq2d 5624 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝐹 “ (dom 𝑓 ∖ {𝑥})) = (𝐹 “ (dom 𝐹 ∖ {𝑥})))
4643, 45eqtrd 2794 . . . . . . . . . . 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 6358 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝑊) → ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) = (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
4938, 48eleq12d 2833 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝑊) → ((𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5049notbid 307 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝑊) → (¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) ↔ ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
5130, 50raleqbidv 3291 . . . . . 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 3291 . . . . 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 272 . . . 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 749 . . 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 20347 . . 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 5139 . 2 ((𝐹𝑋𝑊𝑌) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
5756ancoms 468 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  [wsbc 3576  cdif 3712  {csn 4321   class class class wbr 4804  dom cdm 5266  cima 5269  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  Scalarcsca 16146   ·𝑠 cvsca 16147  0gc0g 16302  LSpanclspn 19173   LIndF clindf 20345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-lindf 20347
This theorem is referenced by:  islinds2  20354  islindf2  20355  lindff  20356  lindfind  20357  f1lindf  20363  lsslindf  20371  matunitlindf  33720
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