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Theorem lindfind 20074
 Description: A linearly independent family 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
lindfind (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))

Proof of Theorem lindfind
Dummy variables 𝑎 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 791 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐸 ∈ dom 𝐹)
2 eldifsn 4287 . . . 4 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
32biimpri 218 . . 3 ((𝐴𝐾𝐴0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
43adantl 482 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5 simpll 789 . . . 4 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐹 LIndF 𝑊)
6 lindfind.l . . . . . . 7 𝐿 = (Scalar‘𝑊)
7 lindfind.k . . . . . . 7 𝐾 = (Base‘𝐿)
86, 7elbasfv 15841 . . . . . 6 (𝐴𝐾𝑊 ∈ V)
98ad2antrl 763 . . . . 5 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝑊 ∈ V)
10 rellindf 20066 . . . . . . 7 Rel LIndF
1110brrelexi 5118 . . . . . 6 (𝐹 LIndF 𝑊𝐹 ∈ V)
1211ad2antrr 761 . . . . 5 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐹 ∈ V)
13 eqid 2621 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
14 lindfind.s . . . . . 6 · = ( ·𝑠𝑊)
15 lindfind.n . . . . . 6 𝑁 = (LSpan‘𝑊)
16 lindfind.z . . . . . 6 0 = (0g𝐿)
1713, 14, 15, 6, 7, 16islindf 20070 . . . . 5 ((𝑊 ∈ V ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))))
189, 12, 17syl2anc 692 . . . 4 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))))
195, 18mpbid 222 . . 3 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒})))))
2019simprd 479 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → ∀𝑒 ∈ dom 𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))
21 fveq2 6148 . . . . . 6 (𝑒 = 𝐸 → (𝐹𝑒) = (𝐹𝐸))
2221oveq2d 6620 . . . . 5 (𝑒 = 𝐸 → (𝑎 · (𝐹𝑒)) = (𝑎 · (𝐹𝐸)))
23 sneq 4158 . . . . . . . 8 (𝑒 = 𝐸 → {𝑒} = {𝐸})
2423difeq2d 3706 . . . . . . 7 (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
2524imaeq2d 5425 . . . . . 6 (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
2625fveq2d 6152 . . . . 5 (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
2722, 26eleq12d 2692 . . . 4 (𝑒 = 𝐸 → ((𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
2827notbid 308 . . 3 (𝑒 = 𝐸 → (¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ ¬ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
29 oveq1 6611 . . . . 5 (𝑎 = 𝐴 → (𝑎 · (𝐹𝐸)) = (𝐴 · (𝐹𝐸)))
3029eleq1d 2683 . . . 4 (𝑎 = 𝐴 → ((𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
3130notbid 308 . . 3 (𝑎 = 𝐴 → (¬ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
3228, 31rspc2va 3307 . 2 (((𝐸 ∈ dom 𝐹𝐴 ∈ (𝐾 ∖ { 0 })) ∧ ∀𝑒 ∈ dom 𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒})))) → ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
331, 4, 20, 32syl21anc 1322 1 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  Vcvv 3186   ∖ cdif 3552  {csn 4148   class class class wbr 4613  dom cdm 5074   “ cima 5077  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  Scalarcsca 15865   ·𝑠 cvsca 15866  0gc0g 16021  LSpanclspn 18890   LIndF clindf 20062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-slot 15785  df-base 15786  df-lindf 20064 This theorem is referenced by:  lindfind2  20076  lindfrn  20079  f1lindf  20080
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