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Theorem lindsind 20204
 Description: A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
lindfind.s · = ( ·𝑠𝑊)
lindfind.n 𝑁 = (LSpan‘𝑊)
lindfind.l 𝐿 = (Scalar‘𝑊)
lindfind.z 0 = (0g𝐿)
lindfind.k 𝐾 = (Base‘𝐿)
Assertion
Ref Expression
lindsind (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))

Proof of Theorem lindsind
Dummy variables 𝑎 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 807 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐸𝐹)
2 eldifsn 4350 . . . 4 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
32biimpri 218 . . 3 ((𝐴𝐾𝐴0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
43adantl 481 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5 elfvdm 6258 . . . . . 6 (𝐹 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
6 eqid 2651 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
7 lindfind.s . . . . . . 7 · = ( ·𝑠𝑊)
8 lindfind.n . . . . . . 7 𝑁 = (LSpan‘𝑊)
9 lindfind.l . . . . . . 7 𝐿 = (Scalar‘𝑊)
10 lindfind.k . . . . . . 7 𝐾 = (Base‘𝐿)
11 lindfind.z . . . . . . 7 0 = (0g𝐿)
126, 7, 8, 9, 10, 11islinds2 20200 . . . . . 6 (𝑊 ∈ dom LIndS → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))))
135, 12syl 17 . . . . 5 (𝐹 ∈ (LIndS‘𝑊) → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))))
1413ibi 256 . . . 4 (𝐹 ∈ (LIndS‘𝑊) → (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))))
1514simprd 478 . . 3 (𝐹 ∈ (LIndS‘𝑊) → ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
1615ad2antrr 762 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
17 oveq2 6698 . . . . 5 (𝑒 = 𝐸 → (𝑎 · 𝑒) = (𝑎 · 𝐸))
18 sneq 4220 . . . . . . 7 (𝑒 = 𝐸 → {𝑒} = {𝐸})
1918difeq2d 3761 . . . . . 6 (𝑒 = 𝐸 → (𝐹 ∖ {𝑒}) = (𝐹 ∖ {𝐸}))
2019fveq2d 6233 . . . . 5 (𝑒 = 𝐸 → (𝑁‘(𝐹 ∖ {𝑒})) = (𝑁‘(𝐹 ∖ {𝐸})))
2117, 20eleq12d 2724 . . . 4 (𝑒 = 𝐸 → ((𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})) ↔ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
2221notbid 307 . . 3 (𝑒 = 𝐸 → (¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})) ↔ ¬ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
23 oveq1 6697 . . . . 5 (𝑎 = 𝐴 → (𝑎 · 𝐸) = (𝐴 · 𝐸))
2423eleq1d 2715 . . . 4 (𝑎 = 𝐴 → ((𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
2524notbid 307 . . 3 (𝑎 = 𝐴 → (¬ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
2622, 25rspc2va 3354 . 2 (((𝐸𝐹𝐴 ∈ (𝐾 ∖ { 0 })) ∧ ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
271, 4, 16, 26syl21anc 1365 1 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941   ∖ cdif 3604   ⊆ wss 3607  {csn 4210  dom cdm 5143  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  LSpanclspn 19019  LIndSclinds 20192 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-lindf 20193  df-linds 20194 This theorem is referenced by: (None)
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