Theorem lbsextlem1 21187

Description: Lemma for lbsext 21192. 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 6928	. . . . 5 ⊢ 𝑉 ∈ V
4	3	elpw2 5343	. . . 4 ⊢ (𝐶 ∈ 𝒫 𝑉 ↔ 𝐶 ⊆ 𝑉)
5	1, 4	sylibr 234	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝑉)
6		lbsext.x	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
7		ssid 4021	. . . 4 ⊢ 𝐶 ⊆ 𝐶
8	6, 7	jctil 519	. . 3 ⊢ (𝜑 → (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
9		sseq2 4025	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐶 ⊆ 𝑧 ↔ 𝐶 ⊆ 𝐶))
10		difeq1 4132	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥}))
11	10	fveq2d 6918	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥})))
12	11	eleq2d 2827	. . . . . . 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 3338	. . . . 5 ⊢ (𝑧 = 𝐶 → (∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
15	9, 14	anbi12d 632	. . . 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 3701	. . 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 4351	1 ⊢ (𝜑 → 𝑆 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  {crab 3436   ∖ cdif 3963   ⊆ wss 3966  ∅c0 4342  𝒫 cpw 4608  {csn 4634  ‘cfv 6569  Basecbs 17254  LSpanclspn 20996  LBasisclbs 21100  LVecclvec 21128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-iota 6522  df-fv 6577

This theorem is referenced by:  lbsextlem4  21190