Theorem lbsextlem1 20420

Description: Lemma for lbsext 20425. 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	⊢ (𝜑 → 𝑊 ∈ LVec)
lbsext.c	⊢ (𝜑 → 𝐶 ⊆ 𝑉)
lbsext.x	⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
lbsext.s	⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}

Assertion

Ref	Expression
lbsextlem1	⊢ (𝜑 → 𝑆 ≠ ∅)

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

Step	Hyp	Ref	Expression
1		lbsext.c	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝑉)
2		lbsext.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
3	2	fvexi 6788	. . . . 5 ⊢ 𝑉 ∈ V
4	3	elpw2 5269	. . . 4 ⊢ (𝐶 ∈ 𝒫 𝑉 ↔ 𝐶 ⊆ 𝑉)
5	1, 4	sylibr 233	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝑉)
6		lbsext.x	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
7		ssid 3943	. . . 4 ⊢ 𝐶 ⊆ 𝐶
8	6, 7	jctil 520	. . 3 ⊢ (𝜑 → (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
9		sseq2 3947	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐶 ⊆ 𝑧 ↔ 𝐶 ⊆ 𝐶))
10		difeq1 4050	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥}))
11	10	fveq2d 6778	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥})))
12	11	eleq2d 2824	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
13	12	notbid 318	. . . . . 6 ⊢ (𝑧 = 𝐶 → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
14	13	raleqbi1dv 3340	. . . . 5 ⊢ (𝑧 = 𝐶 → (∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
15	9, 14	anbi12d 631	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥}))) ↔ (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))))
16		lbsext.s	. . . 4 ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}
17	15, 16	elrab2 3627	. . 3 ⊢ (𝐶 ∈ 𝑆 ↔ (𝐶 ∈ 𝒫 𝑉 ∧ (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))))
18	5, 8, 17	sylanbrc 583	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
19	18	ne0d 4269	1 ⊢ (𝜑 → 𝑆 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3068   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ‘cfv 6433  Basecbs 16912  LSpanclspn 20233  LBasisclbs 20336  LVecclvec 20364

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441

This theorem is referenced by:  lbsextlem4  20423