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Theorem lindfind 20888
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 765 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐸 ∈ dom 𝐹)
2 eldifsn 4711 . . . 4 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
32biimpri 229 . . 3 ((𝐴𝐾𝐴0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
43adantl 482 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5 simpll 763 . . . 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 16532 . . . . . 6 (𝐴𝐾𝑊 ∈ V)
98ad2antrl 724 . . . . 5 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝑊 ∈ V)
10 rellindf 20880 . . . . . . 7 Rel LIndF
1110brrelex1i 5601 . . . . . 6 (𝐹 LIndF 𝑊𝐹 ∈ V)
1211ad2antrr 722 . . . . 5 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐹 ∈ V)
13 eqid 2818 . . . . . 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 20884 . . . . 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 584 . . . 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 233 . . 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 496 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → ∀𝑒 ∈ dom 𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))
21 fveq2 6663 . . . . . 6 (𝑒 = 𝐸 → (𝐹𝑒) = (𝐹𝐸))
2221oveq2d 7161 . . . . 5 (𝑒 = 𝐸 → (𝑎 · (𝐹𝑒)) = (𝑎 · (𝐹𝐸)))
23 sneq 4567 . . . . . . . 8 (𝑒 = 𝐸 → {𝑒} = {𝐸})
2423difeq2d 4096 . . . . . . 7 (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
2524imaeq2d 5922 . . . . . 6 (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
2625fveq2d 6667 . . . . 5 (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
2722, 26eleq12d 2904 . . . 4 (𝑒 = 𝐸 → ((𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
2827notbid 319 . . 3 (𝑒 = 𝐸 → (¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ ¬ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
29 oveq1 7152 . . . . 5 (𝑎 = 𝐴 → (𝑎 · (𝐹𝐸)) = (𝐴 · (𝐹𝐸)))
3029eleq1d 2894 . . . 4 (𝑎 = 𝐴 → ((𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
3130notbid 319 . . 3 (𝑎 = 𝐴 → (¬ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
3228, 31rspc2va 3631 . 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 833 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 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  Vcvv 3492  cdif 3930  {csn 4557   class class class wbr 5057  dom cdm 5548  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  0gc0g 16701  LSpanclspn 19672   LIndF clindf 20876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-slot 16475  df-base 16477  df-lindf 20878
This theorem is referenced by:  lindfind2  20890  lindfrn  20893  f1lindf  20894
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