Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  lindsind Structured version   Visualization version   GIF version

Theorem lindsind 20582
 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 768 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐸𝐹)
2 eldifsn 4677 . . . 4 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
32biimpri 231 . . 3 ((𝐴𝐾𝐴0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
43adantl 485 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5 elfvdm 6690 . . . . . 6 (𝐹 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
6 eqid 2758 . . . . . . 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 20578 . . . . . 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 270 . . . 4 (𝐹 ∈ (LIndS‘𝑊) → (𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))))
1514simprd 499 . . 3 (𝐹 ∈ (LIndS‘𝑊) → ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
1615ad2antrr 725 . 2 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})))
17 oveq2 7158 . . . . 5 (𝑒 = 𝐸 → (𝑎 · 𝑒) = (𝑎 · 𝐸))
18 sneq 4532 . . . . . . 7 (𝑒 = 𝐸 → {𝑒} = {𝐸})
1918difeq2d 4028 . . . . . 6 (𝑒 = 𝐸 → (𝐹 ∖ {𝑒}) = (𝐹 ∖ {𝐸}))
2019fveq2d 6662 . . . . 5 (𝑒 = 𝐸 → (𝑁‘(𝐹 ∖ {𝑒})) = (𝑁‘(𝐹 ∖ {𝐸})))
2117, 20eleq12d 2846 . . . 4 (𝑒 = 𝐸 → ((𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})) ↔ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
2221notbid 321 . . 3 (𝑒 = 𝐸 → (¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒})) ↔ ¬ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
23 oveq1 7157 . . . . 5 (𝑎 = 𝐴 → (𝑎 · 𝐸) = (𝐴 · 𝐸))
2423eleq1d 2836 . . . 4 (𝑎 = 𝐴 → ((𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})) ↔ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
2524notbid 321 . . 3 (𝑎 = 𝐴 → (¬ (𝑎 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})) ↔ ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))))
2622, 25rspc2va 3552 . 2 (((𝐸𝐹𝐴 ∈ (𝐾 ∖ { 0 })) ∧ ∀𝑒𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · 𝑒) ∈ (𝑁‘(𝐹 ∖ {𝑒}))) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
271, 4, 16, 26syl21anc 836 1 (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070   ∖ cdif 3855   ⊆ wss 3858  {csn 4522  dom cdm 5524  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  Scalarcsca 16626   ·𝑠 cvsca 16627  0gc0g 16771  LSpanclspn 19811  LIndSclinds 20570 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-lindf 20571  df-linds 20572 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator