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Theorem lbsextlem1 20664
Description: Lemma for lbsext 20669. The set 𝑆 is the set of all linearly independent sets containing 𝐢; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
Hypotheses
Ref Expression
lbsext.v 𝑉 = (Baseβ€˜π‘Š)
lbsext.j 𝐽 = (LBasisβ€˜π‘Š)
lbsext.n 𝑁 = (LSpanβ€˜π‘Š)
lbsext.w (πœ‘ β†’ π‘Š ∈ LVec)
lbsext.c (πœ‘ β†’ 𝐢 βŠ† 𝑉)
lbsext.x (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐢 Β¬ π‘₯ ∈ (π‘β€˜(𝐢 βˆ– {π‘₯})))
lbsext.s 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐢 βŠ† 𝑧 ∧ βˆ€π‘₯ ∈ 𝑧 Β¬ π‘₯ ∈ (π‘β€˜(𝑧 βˆ– {π‘₯})))}
Assertion
Ref Expression
lbsextlem1 (πœ‘ β†’ 𝑆 β‰  βˆ…)
Distinct variable groups:   π‘₯,𝐽   πœ‘,π‘₯   π‘₯,𝑆   π‘₯,𝑧,𝐢   π‘₯,𝑁,𝑧   π‘₯,𝑉,𝑧   π‘₯,π‘Š
Allowed substitution hints:   πœ‘(𝑧)   𝑆(𝑧)   𝐽(𝑧)   π‘Š(𝑧)

Proof of Theorem lbsextlem1
StepHypRef Expression
1 lbsext.c . . . 4 (πœ‘ β†’ 𝐢 βŠ† 𝑉)
2 lbsext.v . . . . . 6 𝑉 = (Baseβ€˜π‘Š)
32fvexi 6860 . . . . 5 𝑉 ∈ V
43elpw2 5306 . . . 4 (𝐢 ∈ 𝒫 𝑉 ↔ 𝐢 βŠ† 𝑉)
51, 4sylibr 233 . . 3 (πœ‘ β†’ 𝐢 ∈ 𝒫 𝑉)
6 lbsext.x . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐢 Β¬ π‘₯ ∈ (π‘β€˜(𝐢 βˆ– {π‘₯})))
7 ssid 3970 . . . 4 𝐢 βŠ† 𝐢
86, 7jctil 521 . . 3 (πœ‘ β†’ (𝐢 βŠ† 𝐢 ∧ βˆ€π‘₯ ∈ 𝐢 Β¬ π‘₯ ∈ (π‘β€˜(𝐢 βˆ– {π‘₯}))))
9 sseq2 3974 . . . . 5 (𝑧 = 𝐢 β†’ (𝐢 βŠ† 𝑧 ↔ 𝐢 βŠ† 𝐢))
10 difeq1 4079 . . . . . . . . 9 (𝑧 = 𝐢 β†’ (𝑧 βˆ– {π‘₯}) = (𝐢 βˆ– {π‘₯}))
1110fveq2d 6850 . . . . . . . 8 (𝑧 = 𝐢 β†’ (π‘β€˜(𝑧 βˆ– {π‘₯})) = (π‘β€˜(𝐢 βˆ– {π‘₯})))
1211eleq2d 2820 . . . . . . 7 (𝑧 = 𝐢 β†’ (π‘₯ ∈ (π‘β€˜(𝑧 βˆ– {π‘₯})) ↔ π‘₯ ∈ (π‘β€˜(𝐢 βˆ– {π‘₯}))))
1312notbid 318 . . . . . 6 (𝑧 = 𝐢 β†’ (Β¬ π‘₯ ∈ (π‘β€˜(𝑧 βˆ– {π‘₯})) ↔ Β¬ π‘₯ ∈ (π‘β€˜(𝐢 βˆ– {π‘₯}))))
1413raleqbi1dv 3306 . . . . 5 (𝑧 = 𝐢 β†’ (βˆ€π‘₯ ∈ 𝑧 Β¬ π‘₯ ∈ (π‘β€˜(𝑧 βˆ– {π‘₯})) ↔ βˆ€π‘₯ ∈ 𝐢 Β¬ π‘₯ ∈ (π‘β€˜(𝐢 βˆ– {π‘₯}))))
159, 14anbi12d 632 . . . 4 (𝑧 = 𝐢 β†’ ((𝐢 βŠ† 𝑧 ∧ βˆ€π‘₯ ∈ 𝑧 Β¬ π‘₯ ∈ (π‘β€˜(𝑧 βˆ– {π‘₯}))) ↔ (𝐢 βŠ† 𝐢 ∧ βˆ€π‘₯ ∈ 𝐢 Β¬ π‘₯ ∈ (π‘β€˜(𝐢 βˆ– {π‘₯})))))
16 lbsext.s . . . 4 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐢 βŠ† 𝑧 ∧ βˆ€π‘₯ ∈ 𝑧 Β¬ π‘₯ ∈ (π‘β€˜(𝑧 βˆ– {π‘₯})))}
1715, 16elrab2 3652 . . 3 (𝐢 ∈ 𝑆 ↔ (𝐢 ∈ 𝒫 𝑉 ∧ (𝐢 βŠ† 𝐢 ∧ βˆ€π‘₯ ∈ 𝐢 Β¬ π‘₯ ∈ (π‘β€˜(𝐢 βˆ– {π‘₯})))))
185, 8, 17sylanbrc 584 . 2 (πœ‘ β†’ 𝐢 ∈ 𝑆)
1918ne0d 4299 1 (πœ‘ β†’ 𝑆 β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  {crab 3406   βˆ– cdif 3911   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  {csn 4590  β€˜cfv 6500  Basecbs 17091  LSpanclspn 20476  LBasisclbs 20579  LVecclvec 20607
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-ext 2704  ax-sep 5260  ax-nul 5267
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508
This theorem is referenced by:  lbsextlem4  20667
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