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Theorem lindsind 21239
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 768 . 2 (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ 𝐸 ∈ 𝐹)
2 eldifsn 4748 . . . 4 (𝐴 ∈ (𝐾 βˆ– { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 ))
32biimpri 227 . . 3 ((𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 ) β†’ 𝐴 ∈ (𝐾 βˆ– { 0 }))
43adantl 483 . 2 (((𝐹 ∈ (LIndSβ€˜π‘Š) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ 𝐴 ∈ (𝐾 βˆ– { 0 }))
5 elfvdm 6880 . . . . . 6 (𝐹 ∈ (LIndSβ€˜π‘Š) β†’ π‘Š ∈ dom LIndS)
6 eqid 2733 . . . . . . 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 21235 . . . . . 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 497 . . 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 725 . 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 7366 . . . . 5 (𝑒 = 𝐸 β†’ (π‘Ž Β· 𝑒) = (π‘Ž Β· 𝐸))
18 sneq 4597 . . . . . . 7 (𝑒 = 𝐸 β†’ {𝑒} = {𝐸})
1918difeq2d 4083 . . . . . 6 (𝑒 = 𝐸 β†’ (𝐹 βˆ– {𝑒}) = (𝐹 βˆ– {𝐸}))
2019fveq2d 6847 . . . . 5 (𝑒 = 𝐸 β†’ (π‘β€˜(𝐹 βˆ– {𝑒})) = (π‘β€˜(𝐹 βˆ– {𝐸})))
2117, 20eleq12d 2828 . . . 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 7365 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž Β· 𝐸) = (𝐴 Β· 𝐸))
2423eleq1d 2819 . . . 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 3590 . 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 837 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061   βˆ– cdif 3908   βŠ† wss 3911  {csn 4587  dom cdm 5634  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326  LSpanclspn 20447  LIndSclinds 21227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-lindf 21228  df-linds 21229
This theorem is referenced by: (None)
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