Theorem lbsextlem1 20770

Description: Lemma for lbsext 20775. 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 6905	. . . . 5 ⊢ 𝑉 ∈ V
4	3	elpw2 5345	. . . 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 4004	. . . 4 ⊢ 𝐶 ⊆ 𝐶
8	6, 7	jctil 520	. . 3 ⊢ (𝜑 → (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
9		sseq2 4008	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐶 ⊆ 𝑧 ↔ 𝐶 ⊆ 𝐶))
10		difeq1 4115	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥}))
11	10	fveq2d 6895	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥})))
12	11	eleq2d 2819	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
13	12	notbid 317	. . . . . 6 ⊢ (𝑧 = 𝐶 → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))
14	13	raleqbi1dv 3333	. . . . 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 3686	. . 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 4335	1 ⊢ (𝜑 → 𝑆 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  {crab 3432   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628  ‘cfv 6543  Basecbs 17143  LSpanclspn 20581  LBasisclbs 20684  LVecclvec 20712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551

This theorem is referenced by:  lbsextlem4  20773