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Theorem lindsind 21363
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 767 . 2 (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ 𝐸 ∈ 𝐹)
2 eldifsn 4789 . . . 4 (𝐴 ∈ (𝐾 βˆ– { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 ))
32biimpri 227 . . 3 ((𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 ) β†’ 𝐴 ∈ (𝐾 βˆ– { 0 }))
43adantl 482 . 2 (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ 𝐴 ∈ (𝐾 βˆ– { 0 }))
5 elfvdm 6925 . . . . . 6 (𝐹 ∈ (LIndSβ€˜π‘Š) β†’ π‘Š ∈ dom LIndS)
6 eqid 2732 . . . . . . 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 21359 . . . . . 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 266 . . . 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 496 . . 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 724 . 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 7413 . . . . 5 (𝑒 = 𝐸 β†’ (π‘Ž Β· 𝑒) = (π‘Ž Β· 𝐸))
18 sneq 4637 . . . . . . 7 (𝑒 = 𝐸 β†’ {𝑒} = {𝐸})
1918difeq2d 4121 . . . . . 6 (𝑒 = 𝐸 β†’ (𝐹 βˆ– {𝑒}) = (𝐹 βˆ– {𝐸}))
2019fveq2d 6892 . . . . 5 (𝑒 = 𝐸 β†’ (π‘β€˜(𝐹 βˆ– {𝑒})) = (π‘β€˜(𝐹 βˆ– {𝐸})))
2117, 20eleq12d 2827 . . . 4 (𝑒 = 𝐸 β†’ ((π‘Ž Β· 𝑒) ∈ (π‘β€˜(𝐹 βˆ– {𝑒})) ↔ (π‘Ž Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸}))))
2221notbid 317 . . 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 7412 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž Β· 𝐸) = (𝐴 Β· 𝐸))
2423eleq1d 2818 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘Ž Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸})) ↔ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐹 βˆ– {𝐸}))))
2524notbid 317 . . 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 3622 . 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 836 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   βˆ– cdif 3944   βŠ† wss 3947  {csn 4627  dom cdm 5675  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  LSpanclspn 20574  LIndSclinds 21351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-lindf 21352  df-linds 21353
This theorem is referenced by: (None)
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