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Theorem lindsind 19975
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 788 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐸𝐹)
2 eldifsn 4260 . . . 4 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
32biimpri 217 . . 3 ((𝐴𝐾𝐴0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
43adantl 481 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5 elfvdm 6130 . . . . . 6 (𝐹 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
6 eqid 2610 . . . . . . 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 19971 . . . . . 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 255 . . . 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 758 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
17 oveq2 6557 . . . . 5 (𝑒 = 𝐸 → (𝑎 · 𝑒) = (𝑎 · 𝐸))
18 sneq 4135 . . . . . . 7 (𝑒 = 𝐸 → {𝑒} = {𝐸})
1918difeq2d 3690 . . . . . 6 (𝑒 = 𝐸 → (𝐹 ∖ {𝑒}) = (𝐹 ∖ {𝐸}))
2019fveq2d 6107 . . . . 5 (𝑒 = 𝐸 → (𝑁‘(𝐹 ∖ {𝑒})) = (𝑁‘(𝐹 ∖ {𝐸})))
2117, 20eleq12d 2682 . . . 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 6556 . . . . 5 (𝑎 = 𝐴 → (𝑎 · 𝐸) = (𝐴 · 𝐸))
2423eleq1d 2672 . . . 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 3294 . 2 (((𝐸𝐹𝐴 ∈ (𝐾 ∖ { 0 })) ∧ ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
271, 4, 16, 26syl21anc 1317 1 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  cdif 3537  wss 3540  {csn 4125  dom cdm 5038  cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  LSpanclspn 18792  LIndSclinds 19963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-lindf 19964  df-linds 19965
This theorem is referenced by: (None)
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