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Theorem lindsind 21592
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 Β· = ( ·𝑠 β€˜π‘Š)
lindfind.n 𝑁 = (LSpanβ€˜π‘Š)
lindfind.l 𝐿 = (Scalarβ€˜π‘Š)
lindfind.z 0 = (0gβ€˜πΏ)
lindfind.k 𝐾 = (Baseβ€˜πΏ)
Assertion
Ref Expression
lindsind (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸})))

Proof of Theorem lindsind
Dummy variables π‘Ž 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 766 . 2 (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ 𝐸 ∈ 𝐹)
2 eldifsn 4790 . . . 4 (𝐴 ∈ (𝐾 βˆ– { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 ))
32biimpri 227 . . 3 ((𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 ) β†’ 𝐴 ∈ (𝐾 βˆ– { 0 }))
43adantl 481 . 2 (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ 𝐴 ∈ (𝐾 βˆ– { 0 }))
5 elfvdm 6928 . . . . . 6 (𝐹 ∈ (LIndSβ€˜π‘Š) β†’ π‘Š ∈ dom LIndS)
6 eqid 2731 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
7 lindfind.s . . . . . . 7 Β· = ( ·𝑠 β€˜π‘Š)
8 lindfind.n . . . . . . 7 𝑁 = (LSpanβ€˜π‘Š)
9 lindfind.l . . . . . . 7 𝐿 = (Scalarβ€˜π‘Š)
10 lindfind.k . . . . . . 7 𝐾 = (Baseβ€˜πΏ)
11 lindfind.z . . . . . . 7 0 = (0gβ€˜πΏ)
126, 7, 8, 9, 10, 11islinds2 21588 . . . . . 6 (π‘Š ∈ dom LIndS β†’ (𝐹 ∈ (LIndSβ€˜π‘Š) ↔ (𝐹 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘’ ∈ 𝐹 βˆ€π‘Ž ∈ (𝐾 βˆ– { 0 }) Β¬ (π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒})))))
135, 12syl 17 . . . . 5 (𝐹 ∈ (LIndSβ€˜π‘Š) β†’ (𝐹 ∈ (LIndSβ€˜π‘Š) ↔ (𝐹 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘’ ∈ 𝐹 βˆ€π‘Ž ∈ (𝐾 βˆ– { 0 }) Β¬ (π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒})))))
1413ibi 267 . . . 4 (𝐹 ∈ (LIndSβ€˜π‘Š) β†’ (𝐹 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘’ ∈ 𝐹 βˆ€π‘Ž ∈ (𝐾 βˆ– { 0 }) Β¬ (π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒}))))
1514simprd 495 . . 3 (𝐹 ∈ (LIndSβ€˜π‘Š) β†’ βˆ€π‘’ ∈ 𝐹 βˆ€π‘Ž ∈ (𝐾 βˆ– { 0 }) Β¬ (π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒})))
1615ad2antrr 723 . 2 (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ βˆ€π‘’ ∈ 𝐹 βˆ€π‘Ž ∈ (𝐾 βˆ– { 0 }) Β¬ (π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒})))
17 oveq2 7420 . . . . 5 (𝑒 = 𝐸 β†’ (π‘Ž Β· 𝑒) = (π‘Ž Β· 𝐸))
18 sneq 4638 . . . . . . 7 (𝑒 = 𝐸 β†’ {𝑒} = {𝐸})
1918difeq2d 4122 . . . . . 6 (𝑒 = 𝐸 β†’ (𝐹 βˆ– {𝑒}) = (𝐹 βˆ– {𝐸}))
2019fveq2d 6895 . . . . 5 (𝑒 = 𝐸 β†’ (π‘β€˜(𝐹 βˆ– {𝑒})) = (π‘β€˜(𝐹 βˆ– {𝐸})))
2117, 20eleq12d 2826 . . . 4 (𝑒 = 𝐸 β†’ ((π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒})) ↔ (π‘Ž Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸}))))
2221notbid 318 . . 3 (𝑒 = 𝐸 β†’ (Β¬ (π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒})) ↔ Β¬ (π‘Ž Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸}))))
23 oveq1 7419 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž Β· 𝐸) = (𝐴 Β· 𝐸))
2423eleq1d 2817 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘Ž Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸})) ↔ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸}))))
2524notbid 318 . . 3 (π‘Ž = 𝐴 β†’ (Β¬ (π‘Ž Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸})) ↔ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸}))))
2622, 25rspc2va 3623 . 2 (((𝐸 ∈ 𝐹 ∧ 𝐴 ∈ (𝐾 βˆ– { 0 })) ∧ βˆ€π‘’ ∈ 𝐹 βˆ€π‘Ž ∈ (𝐾 βˆ– { 0 }) Β¬ (π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒}))) β†’ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸})))
271, 4, 16, 26syl21anc 835 1 (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060   βˆ– cdif 3945   βŠ† wss 3948  {csn 4628  dom cdm 5676  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  Scalarcsca 17205   ·𝑠 cvsca 17206  0gc0g 17390  LSpanclspn 20727  LIndSclinds 21580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-lindf 21581  df-linds 21582
This theorem is referenced by: (None)
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