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Theorem lbsextlem1 20003
 Description: Lemma for lbsext 20008. The set 𝑆 is the set of all linearly independent sets containing 𝐶; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
Hypotheses
Ref Expression
lbsext.v 𝑉 = (Base‘𝑊)
lbsext.j 𝐽 = (LBasis‘𝑊)
lbsext.n 𝑁 = (LSpan‘𝑊)
lbsext.w (𝜑𝑊 ∈ LVec)
lbsext.c (𝜑𝐶𝑉)
lbsext.x (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
lbsext.s 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}
Assertion
Ref Expression
lbsextlem1 (𝜑𝑆 ≠ ∅)
Distinct variable groups:   𝑥,𝐽   𝜑,𝑥   𝑥,𝑆   𝑥,𝑧,𝐶   𝑥,𝑁,𝑧   𝑥,𝑉,𝑧   𝑥,𝑊
Allowed substitution hints:   𝜑(𝑧)   𝑆(𝑧)   𝐽(𝑧)   𝑊(𝑧)

Proof of Theorem lbsextlem1
StepHypRef Expression
1 lbsext.c . . . 4 (𝜑𝐶𝑉)
2 lbsext.v . . . . . 6 𝑉 = (Base‘𝑊)
32fvexi 6676 . . . . 5 𝑉 ∈ V
43elpw2 5218 . . . 4 (𝐶 ∈ 𝒫 𝑉𝐶𝑉)
51, 4sylibr 237 . . 3 (𝜑𝐶 ∈ 𝒫 𝑉)
6 lbsext.x . . . 4 (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
7 ssid 3916 . . . 4 𝐶𝐶
86, 7jctil 523 . . 3 (𝜑 → (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
9 sseq2 3920 . . . . 5 (𝑧 = 𝐶 → (𝐶𝑧𝐶𝐶))
10 difeq1 4023 . . . . . . . . 9 (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥}))
1110fveq2d 6666 . . . . . . . 8 (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥})))
1211eleq2d 2837 . . . . . . 7 (𝑧 = 𝐶 → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1312notbid 321 . . . . . 6 (𝑧 = 𝐶 → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1413raleqbi1dv 3321 . . . . 5 (𝑧 = 𝐶 → (∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
159, 14anbi12d 633 . . . 4 (𝑧 = 𝐶 → ((𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥}))) ↔ (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))))
16 lbsext.s . . . 4 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}
1715, 16elrab2 3607 . . 3 (𝐶𝑆 ↔ (𝐶 ∈ 𝒫 𝑉 ∧ (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))))
185, 8, 17sylanbrc 586 . 2 (𝜑𝐶𝑆)
1918ne0d 4236 1 (𝜑𝑆 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  {crab 3074   ∖ cdif 3857   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  {csn 4525  ‘cfv 6339  Basecbs 16546  LSpanclspn 19816  LBasisclbs 19919  LVecclvec 19947 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-iota 6298  df-fv 6347 This theorem is referenced by:  lbsextlem4  20006
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