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Theorem lindfind 20203
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 807 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐸 ∈ dom 𝐹)
2 eldifsn 4350 . . . 4 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
32biimpri 218 . . 3 ((𝐴𝐾𝐴0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
43adantl 481 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5 simpll 805 . . . 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 15967 . . . . . 6 (𝐴𝐾𝑊 ∈ V)
98ad2antrl 764 . . . . 5 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝑊 ∈ V)
10 rellindf 20195 . . . . . . 7 Rel LIndF
1110brrelexi 5192 . . . . . 6 (𝐹 LIndF 𝑊𝐹 ∈ V)
1211ad2antrr 762 . . . . 5 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐹 ∈ V)
13 eqid 2651 . . . . . 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 20199 . . . . 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 694 . . . 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 478 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → ∀𝑒 ∈ dom 𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))
21 fveq2 6229 . . . . . 6 (𝑒 = 𝐸 → (𝐹𝑒) = (𝐹𝐸))
2221oveq2d 6706 . . . . 5 (𝑒 = 𝐸 → (𝑎 · (𝐹𝑒)) = (𝑎 · (𝐹𝐸)))
23 sneq 4220 . . . . . . . 8 (𝑒 = 𝐸 → {𝑒} = {𝐸})
2423difeq2d 3761 . . . . . . 7 (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
2524imaeq2d 5501 . . . . . 6 (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
2625fveq2d 6233 . . . . 5 (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
2722, 26eleq12d 2724 . . . 4 (𝑒 = 𝐸 → ((𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
2827notbid 307 . . 3 (𝑒 = 𝐸 → (¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ ¬ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
29 oveq1 6697 . . . . 5 (𝑎 = 𝐴 → (𝑎 · (𝐹𝐸)) = (𝐴 · (𝐹𝐸)))
3029eleq1d 2715 . . . 4 (𝑎 = 𝐴 → ((𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
3130notbid 307 . . 3 (𝑎 = 𝐴 → (¬ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
3228, 31rspc2va 3354 . 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 1365 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 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  {csn 4210   class class class wbr 4685  dom cdm 5143  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  LSpanclspn 19019   LIndF clindf 20191
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
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-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-fv 5934  df-ov 6693  df-slot 15908  df-base 15910  df-lindf 20193
This theorem is referenced by:  lindfind2  20205  lindfrn  20208  f1lindf  20209
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