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Theorem lbsind 20926
Description: A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)
Hypotheses
Ref Expression
lbsss.v 𝑉 = (Baseβ€˜π‘Š)
lbsss.j 𝐽 = (LBasisβ€˜π‘Š)
lbssp.n 𝑁 = (LSpanβ€˜π‘Š)
lbsind.f 𝐹 = (Scalarβ€˜π‘Š)
lbsind.s Β· = ( ·𝑠 β€˜π‘Š)
lbsind.k 𝐾 = (Baseβ€˜πΉ)
lbsind.z 0 = (0gβ€˜πΉ)
Assertion
Ref Expression
lbsind (((𝐡 ∈ 𝐽 ∧ 𝐸 ∈ 𝐡) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸})))

Proof of Theorem lbsind
Dummy variables 𝑦 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4785 . 2 (𝐴 ∈ (𝐾 βˆ– { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 ))
2 elfvdm 6921 . . . . . . . 8 (𝐡 ∈ (LBasisβ€˜π‘Š) β†’ π‘Š ∈ dom LBasis)
3 lbsss.j . . . . . . . 8 𝐽 = (LBasisβ€˜π‘Š)
42, 3eleq2s 2845 . . . . . . 7 (𝐡 ∈ 𝐽 β†’ π‘Š ∈ dom LBasis)
5 lbsss.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
6 lbsind.f . . . . . . . 8 𝐹 = (Scalarβ€˜π‘Š)
7 lbsind.s . . . . . . . 8 Β· = ( ·𝑠 β€˜π‘Š)
8 lbsind.k . . . . . . . 8 𝐾 = (Baseβ€˜πΉ)
9 lbssp.n . . . . . . . 8 𝑁 = (LSpanβ€˜π‘Š)
10 lbsind.z . . . . . . . 8 0 = (0gβ€˜πΉ)
115, 6, 7, 8, 3, 9, 10islbs 20922 . . . . . . 7 (π‘Š ∈ dom LBasis β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
124, 11syl 17 . . . . . 6 (𝐡 ∈ 𝐽 β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
1312ibi 267 . . . . 5 (𝐡 ∈ 𝐽 β†’ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))))
1413simp3d 1141 . . . 4 (𝐡 ∈ 𝐽 β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))
15 oveq2 7412 . . . . . . 7 (π‘₯ = 𝐸 β†’ (𝑦 Β· π‘₯) = (𝑦 Β· 𝐸))
16 sneq 4633 . . . . . . . . 9 (π‘₯ = 𝐸 β†’ {π‘₯} = {𝐸})
1716difeq2d 4117 . . . . . . . 8 (π‘₯ = 𝐸 β†’ (𝐡 βˆ– {π‘₯}) = (𝐡 βˆ– {𝐸}))
1817fveq2d 6888 . . . . . . 7 (π‘₯ = 𝐸 β†’ (π‘β€˜(𝐡 βˆ– {π‘₯})) = (π‘β€˜(𝐡 βˆ– {𝐸})))
1915, 18eleq12d 2821 . . . . . 6 (π‘₯ = 𝐸 β†’ ((𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})) ↔ (𝑦 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸}))))
2019notbid 318 . . . . 5 (π‘₯ = 𝐸 β†’ (Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})) ↔ Β¬ (𝑦 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸}))))
21 oveq1 7411 . . . . . . 7 (𝑦 = 𝐴 β†’ (𝑦 Β· 𝐸) = (𝐴 Β· 𝐸))
2221eleq1d 2812 . . . . . 6 (𝑦 = 𝐴 β†’ ((𝑦 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸})) ↔ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸}))))
2322notbid 318 . . . . 5 (𝑦 = 𝐴 β†’ (Β¬ (𝑦 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸})) ↔ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸}))))
2420, 23rspc2v 3617 . . . 4 ((𝐸 ∈ 𝐡 ∧ 𝐴 ∈ (𝐾 βˆ– { 0 })) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})) β†’ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸}))))
2514, 24syl5com 31 . . 3 (𝐡 ∈ 𝐽 β†’ ((𝐸 ∈ 𝐡 ∧ 𝐴 ∈ (𝐾 βˆ– { 0 })) β†’ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸}))))
2625impl 455 . 2 (((𝐡 ∈ 𝐽 ∧ 𝐸 ∈ 𝐡) ∧ 𝐴 ∈ (𝐾 βˆ– { 0 })) β†’ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸})))
271, 26sylan2br 594 1 (((𝐡 ∈ 𝐽 ∧ 𝐸 ∈ 𝐡) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0 )) β†’ Β¬ (𝐴 Β· 𝐸) ∈ (π‘β€˜(𝐡 βˆ– {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055   βˆ– cdif 3940   βŠ† wss 3943  {csn 4623  dom cdm 5669  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  Scalarcsca 17207   ·𝑠 cvsca 17208  0gc0g 17392  LSpanclspn 20816  LBasisclbs 20920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-lbs 20921
This theorem is referenced by:  lbsind2  20927
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