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Theorem lindfind 21836
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 769 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐸 ∈ dom 𝐹)
2 eldifsn 4786 . . . 4 (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴𝐾𝐴0 ))
32biimpri 228 . . 3 ((𝐴𝐾𝐴0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
43adantl 481 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5 simpll 767 . . . 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 17253 . . . . . 6 (𝐴𝐾𝑊 ∈ V)
98ad2antrl 728 . . . . 5 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝑊 ∈ V)
10 rellindf 21828 . . . . . . 7 Rel LIndF
1110brrelex1i 5741 . . . . . 6 (𝐹 LIndF 𝑊𝐹 ∈ V)
1211ad2antrr 726 . . . . 5 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → 𝐹 ∈ V)
13 eqid 2737 . . . . . 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 21832 . . . . 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 232 . . 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 495 . 2 (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → ∀𝑒 ∈ dom 𝐹𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))
21 fveq2 6906 . . . . . 6 (𝑒 = 𝐸 → (𝐹𝑒) = (𝐹𝐸))
2221oveq2d 7447 . . . . 5 (𝑒 = 𝐸 → (𝑎 · (𝐹𝑒)) = (𝑎 · (𝐹𝐸)))
23 sneq 4636 . . . . . . . 8 (𝑒 = 𝐸 → {𝑒} = {𝐸})
2423difeq2d 4126 . . . . . . 7 (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
2524imaeq2d 6078 . . . . . 6 (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
2625fveq2d 6910 . . . . 5 (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
2722, 26eleq12d 2835 . . . 4 (𝑒 = 𝐸 → ((𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
2827notbid 318 . . 3 (𝑒 = 𝐸 → (¬ (𝑎 · (𝐹𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ ¬ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
29 oveq1 7438 . . . . 5 (𝑎 = 𝐴 → (𝑎 · (𝐹𝐸)) = (𝐴 · (𝐹𝐸)))
3029eleq1d 2826 . . . 4 (𝑎 = 𝐴 → ((𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
3130notbid 318 . . 3 (𝑎 = 𝐴 → (¬ (𝑎 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
3228, 31rspc2va 3634 . 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 838 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 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cdif 3948  {csn 4626   class class class wbr 5143  dom cdm 5685  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  LSpanclspn 20969   LIndF clindf 21824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-slot 17219  df-ndx 17231  df-base 17248  df-lindf 21826
This theorem is referenced by:  lindfind2  21838  lindfrn  21841  f1lindf  21842
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