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

Theorem lbsextlem1 19072
Description: Lemma for lbsext 19077. 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 6160 . . . . . 6 (Base‘𝑊) ∈ V
42, 3eqeltri 2700 . . . . 5 𝑉 ∈ V
54elpw2 4793 . . . 4 (𝐶 ∈ 𝒫 𝑉𝐶𝑉)
61, 5sylibr 224 . . 3 (𝜑𝐶 ∈ 𝒫 𝑉)
7 lbsext.x . . . 4 (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
8 ssid 3608 . . . 4 𝐶𝐶
97, 8jctil 559 . . 3 (𝜑 → (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
10 sseq2 3611 . . . . 5 (𝑧 = 𝐶 → (𝐶𝑧𝐶𝐶))
11 difeq1 3704 . . . . . . . . 9 (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥}))
1211fveq2d 6154 . . . . . . . 8 (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥})))
1312eleq2d 2689 . . . . . . 7 (𝑧 = 𝐶 → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1413notbid 308 . . . . . 6 (𝑧 = 𝐶 → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1514raleqbi1dv 3140 . . . . 5 (𝑧 = 𝐶 → (∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1610, 15anbi12d 746 . . . 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 3353 . . 3 (𝐶𝑆 ↔ (𝐶 ∈ 𝒫 𝑉 ∧ (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))))
196, 9, 18sylanbrc 697 . 2 (𝜑𝐶𝑆)
20 ne0i 3902 . 2 (𝐶𝑆𝑆 ≠ ∅)
2119, 20syl 17 1 (𝜑𝑆 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  {crab 2916  Vcvv 3191  cdif 3557  wss 3560  c0 3896  𝒫 cpw 4135  {csn 4153  cfv 5850  Basecbs 15776  LSpanclspn 18885  LBasisclbs 18988  LVecclvec 19016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858
This theorem is referenced by:  lbsextlem4  19075
  Copyright terms: Public domain W3C validator