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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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