Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  lbsextlem1 Structured version   Visualization version   GIF version

Theorem lbsextlem1 18979
 Description: Lemma for lbsext 18984. 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)
3 fvex 6113 . . . . . 6 (Base‘𝑊) ∈ V
42, 3eqeltri 2684 . . . . 5 𝑉 ∈ V
54elpw2 4755 . . . 4 (𝐶 ∈ 𝒫 𝑉𝐶𝑉)
61, 5sylibr 223 . . 3 (𝜑𝐶 ∈ 𝒫 𝑉)
7 lbsext.x . . . 4 (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
8 ssid 3587 . . . 4 𝐶𝐶
97, 8jctil 558 . . 3 (𝜑 → (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
10 sseq2 3590 . . . . 5 (𝑧 = 𝐶 → (𝐶𝑧𝐶𝐶))
11 difeq1 3683 . . . . . . . . 9 (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥}))
1211fveq2d 6107 . . . . . . . 8 (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥})))
1312eleq2d 2673 . . . . . . 7 (𝑧 = 𝐶 → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1413notbid 307 . . . . . 6 (𝑧 = 𝐶 → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1514raleqbi1dv 3123 . . . . 5 (𝑧 = 𝐶 → (∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1610, 15anbi12d 743 . . . 4 (𝑧 = 𝐶 → ((𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥}))) ↔ (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))))
17 lbsext.s . . . 4 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}
1816, 17elrab2 3333 . . 3 (𝐶𝑆 ↔ (𝐶 ∈ 𝒫 𝑉 ∧ (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))))
196, 9, 18sylanbrc 695 . 2 (𝜑𝐶𝑆)
20 ne0i 3880 . 2 (𝐶𝑆𝑆 ≠ ∅)
2119, 20syl 17 1 (𝜑𝑆 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ‘cfv 5804  Basecbs 15695  LSpanclspn 18792  LBasisclbs 18895  LVecclvec 18923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  lbsextlem4  18982
 Copyright terms: Public domain W3C validator