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Theorem lbsextlem1 19380
Description: Lemma for lbsext 19385. 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 6363 . . . . . 6 (Base‘𝑊) ∈ V
42, 3eqeltri 2835 . . . . 5 𝑉 ∈ V
54elpw2 4977 . . . 4 (𝐶 ∈ 𝒫 𝑉𝐶𝑉)
61, 5sylibr 224 . . 3 (𝜑𝐶 ∈ 𝒫 𝑉)
7 lbsext.x . . . 4 (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
8 ssid 3765 . . . 4 𝐶𝐶
97, 8jctil 561 . . 3 (𝜑 → (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
10 sseq2 3768 . . . . 5 (𝑧 = 𝐶 → (𝐶𝑧𝐶𝐶))
11 difeq1 3864 . . . . . . . . 9 (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥}))
1211fveq2d 6357 . . . . . . . 8 (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥})))
1312eleq2d 2825 . . . . . . 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 3285 . . . . 5 (𝑧 = 𝐶 → (∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
1610, 15anbi12d 749 . . . 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 3507 . . 3 (𝐶𝑆 ↔ (𝐶 ∈ 𝒫 𝑉 ∧ (𝐶𝐶 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))))
196, 9, 18sylanbrc 701 . 2 (𝜑𝐶𝑆)
20 ne0i 4064 . 2 (𝐶𝑆𝑆 ≠ ∅)
2119, 20syl 17 1 (𝜑𝑆 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  {crab 3054  Vcvv 3340  cdif 3712  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321  cfv 6049  Basecbs 16079  LSpanclspn 19193  LBasisclbs 19296  LVecclvec 19324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  lbsextlem4  19383
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