Theorem lbsextlem2 21294

Description: Lemma for lbsext 21298. Since 𝐴 is a chain (actually, we only need it to be closed under binary union), the union 𝑇 of the spans of each individual element of 𝐴 is a subspace, and it contains all of ∪ 𝐴 (except for our target vector 𝑥- we are trying to make 𝑥 a linear combination of all the other vectors in some set from 𝐴). (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})))}
lbsext.p	⊢ 𝑃 = (LSubSp‘𝑊)
lbsext.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)
lbsext.z	⊢ (𝜑 → 𝐴 ≠ ∅)
lbsext.r	⊢ (𝜑 → [⊊] Or 𝐴)
lbsext.t	⊢ 𝑇 = ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))

Assertion

Ref	Expression
lbsextlem2	⊢ (𝜑 → (𝑇 ∈ 𝑃 ∧ (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑢,𝜑   𝑢,𝑆,𝑥   𝑥,𝑧,𝐶   𝑧,𝑢,𝑁,𝑥   𝑢,𝑉,𝑥,𝑧   𝑢,𝑊,𝑥   𝑢,𝐴,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧)   𝐶(𝑢)   𝑃(𝑥, 𝑧, 𝑢)   𝑆(𝑧)   𝑇(𝑥, 𝑧, 𝑢)   𝐽(𝑧, 𝑢)   𝑊(𝑧)

Proof of Theorem lbsextlem2

Dummy variables 𝑚 𝑛 𝑟 𝑣 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2763	. . 3 ⊢ (𝜑 → (Scalar‘𝑊) = (Scalar‘𝑊))
2		eqidd 2763	. . 3 ⊢ (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
3		lbsext.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑊))
5		eqidd 2763	. . 3 ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝑊))
6		eqidd 2763	. . 3 ⊢ (𝜑 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊))
7		lbsext.p	. . . 4 ⊢ 𝑃 = (LSubSp‘𝑊)
8	7	a1i 11	. . 3 ⊢ (𝜑 → 𝑃 = (LSubSp‘𝑊))
9		lbsext.t	. . . 4 ⊢ 𝑇 = ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))
10		lbsext.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec)
11		lveclmod 21238	. . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
13		lbsext.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
14		lbsext.s	. . . . . . . . . . . 12 ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}
15	14	ssrab3 4035	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ 𝒫 𝑉
16	13, 15	sstrdi 3948	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑉)
17	16	sselda 3936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝒫 𝑉)
18	17	elpwid 4570	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → 𝑢 ⊆ 𝑉)
19	18	ssdifssd 4100	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ 𝑉)
20		lbsext.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
21	3, 20	lspssv 21115	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
22	12, 19, 21	syl2an2r 697	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
23	22	ralrimiva 3156	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
24		iunss 5008	. . . . 5 ⊢ (∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉 ↔ ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
25	23, 24	sylibr 237	. . . 4 ⊢ (𝜑 → ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
26	9, 25	eqsstrid 3974	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑉)
27	9	a1i 11	. . . 4 ⊢ (𝜑 → 𝑇 = ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
28		lbsext.z	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ ∅)
29	3, 7, 20	lspcl 21108	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
30	12, 19, 29	syl2an2r 697	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
31	7	lssn0 21072	. . . . . . . 8 ⊢ ((𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃 → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
32	30, 31	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
33	32	ralrimiva 3156	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
34		r19.2z 4459	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅) → ∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
35	28, 33, 34	syl2anc 595	. . . . 5 ⊢ (𝜑 → ∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
36		iunn0 5030	. . . . 5 ⊢ (∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅ ↔ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
37	35, 36	sylib 221	. . . 4 ⊢ (𝜑 → ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
38	27, 37	eqnetrd 3024	. . 3 ⊢ (𝜑 → 𝑇 ≠ ∅)
39	9	eleq2i 2854	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝑇 ↔ 𝑣 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
40		eliun 4959	. . . . . . . . 9 ⊢ (𝑣 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
41		difeq1 4073	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∖ {𝑥}) = (𝑚 ∖ {𝑥}))
42	41	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑚 ∖ {𝑥})))
43	42	eleq2d 2848	. . . . . . . . . 10 ⊢ (𝑢 = 𝑚 → (𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥}))))
44	43	cbvrexvw 3243	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
45	39, 40, 44	3bitri 300	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑇 ↔ ∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
46	9	eleq2i 2854	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑇 ↔ 𝑤 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
47		eliun 4959	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
48		difeq1 4073	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑛 → (𝑢 ∖ {𝑥}) = (𝑛 ∖ {𝑥}))
49	48	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑛 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑛 ∖ {𝑥})))
50	49	eleq2d 2848	. . . . . . . . . 10 ⊢ (𝑢 = 𝑛 → (𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
51	50	cbvrexvw 3243	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
52	46, 47, 51	3bitri 300	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑇 ↔ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
53	45, 52	anbi12i 639	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ (∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
54		reeanv 3236	. . . . . . 7 ⊢ (∃𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) ↔ (∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
55	53, 54	bitr4i 281	. . . . . 6 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ ∃𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
56		simp1l 1215	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝜑)
57		lbsext.r	. . . . . . . . . . . 12 ⊢ (𝜑 → [⊊] Or 𝐴)
58	56, 57	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → [⊊] Or 𝐴)
59		simp2 1154	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴))
60		sorpssun 7729	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴)) → (𝑚 ∪ 𝑛) ∈ 𝐴)
61	58, 59, 60	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∪ 𝑛) ∈ 𝐴)
62	56, 12	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑊 ∈ LMod)
63		elssuni 4903	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∪ 𝑛) ∈ 𝐴 → (𝑚 ∪ 𝑛) ⊆ ∪ 𝐴)
64	61, 63	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∪ 𝑛) ⊆ ∪ 𝐴)
65		sspwuni 5065	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ 𝒫 𝑉 ↔ ∪ 𝐴 ⊆ 𝑉)
66	16, 65	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∪ 𝐴 ⊆ 𝑉)
67	56, 66	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ∪ 𝐴 ⊆ 𝑉)
68	64, 67	sstrd 3946	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∪ 𝑛) ⊆ 𝑉)
69	68	ssdifssd 4100	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑚 ∪ 𝑛) ∖ {𝑥}) ⊆ 𝑉)
70	3, 7, 20	lspcl 21108	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ ((𝑚 ∪ 𝑛) ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})) ∈ 𝑃)
71	62, 69, 70	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})) ∈ 𝑃)
72		simp1r 1216	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
73		ssun1 4130	. . . . . . . . . . . . . 14 ⊢ 𝑚 ⊆ (𝑚 ∪ 𝑛)
74		ssdif 4097	. . . . . . . . . . . . . 14 ⊢ (𝑚 ⊆ (𝑚 ∪ 𝑛) → (𝑚 ∖ {𝑥}) ⊆ ((𝑚 ∪ 𝑛) ∖ {𝑥}))
75	73, 74	mp1i 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∖ {𝑥}) ⊆ ((𝑚 ∪ 𝑛) ∖ {𝑥}))
76	3, 20	lspss 21116	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ ((𝑚 ∪ 𝑛) ∖ {𝑥}) ⊆ 𝑉 ∧ (𝑚 ∖ {𝑥}) ⊆ ((𝑚 ∪ 𝑛) ∖ {𝑥})) → (𝑁‘(𝑚 ∖ {𝑥})) ⊆ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
77	62, 69, 75, 76	syl3anc 1397	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘(𝑚 ∖ {𝑥})) ⊆ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
78		simp3l 1219	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
79	77, 78	sseldd 3937	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑣 ∈ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
80		ssun2 4131	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ (𝑚 ∪ 𝑛)
81		ssdif 4097	. . . . . . . . . . . . . 14 ⊢ (𝑛 ⊆ (𝑚 ∪ 𝑛) → (𝑛 ∖ {𝑥}) ⊆ ((𝑚 ∪ 𝑛) ∖ {𝑥}))
82	80, 81	mp1i 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑛 ∖ {𝑥}) ⊆ ((𝑚 ∪ 𝑛) ∖ {𝑥}))
83	3, 20	lspss 21116	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ ((𝑚 ∪ 𝑛) ∖ {𝑥}) ⊆ 𝑉 ∧ (𝑛 ∖ {𝑥}) ⊆ ((𝑚 ∪ 𝑛) ∖ {𝑥})) → (𝑁‘(𝑛 ∖ {𝑥})) ⊆ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
84	62, 69, 82, 83	syl3anc 1397	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘(𝑛 ∖ {𝑥})) ⊆ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
85		simp3r 1220	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
86	84, 85	sseldd 3937	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑤 ∈ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
87		eqid 2762	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
88		eqid 2762	. . . . . . . . . . . 12 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
89		eqid 2762	. . . . . . . . . . . 12 ⊢ (+g‘𝑊) = (+g‘𝑊)
90		eqid 2762	. . . . . . . . . . . 12 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
91	87, 88, 89, 90, 7	lsscl 21074	. . . . . . . . . . 11 ⊢ (((𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})) ∈ 𝑃 ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑣 ∈ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
92	71, 72, 79, 86, 91	syl13anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
93		difeq1 4073	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑚 ∪ 𝑛) → (𝑢 ∖ {𝑥}) = ((𝑚 ∪ 𝑛) ∖ {𝑥}))
94	93	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑚 ∪ 𝑛) → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
95	94	eliuni 4961	. . . . . . . . . 10 ⊢ (((𝑚 ∪ 𝑛) ∈ 𝐴 ∧ ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥}))) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
96	61, 92, 95	syl2anc 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
97	96, 9	eleqtrrdi 2873	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇)
98	97	3expia 1138	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴)) → ((𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇))
99	98	rexlimdvva 3221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) → (∃𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇))
100	55, 99	biimtrid 245	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) → ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇))
101	100	exp4b 435	. . . 4 ⊢ (𝜑 → (𝑟 ∈ (Base‘(Scalar‘𝑊)) → (𝑣 ∈ 𝑇 → (𝑤 ∈ 𝑇 → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇))))
102	101	3imp2 1367	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇)) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇)
103	1, 2, 4, 5, 6, 8, 26, 38, 102	islssd 21067	. 2 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
104		eldifi 4084	. . . . . . 7 ⊢ (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → 𝑦 ∈ ∪ 𝐴)
105	104	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → 𝑦 ∈ ∪ 𝐴)
106		eldifn 4085	. . . . . . . . . 10 ⊢ (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → ¬ 𝑦 ∈ {𝑥})
107	106	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → ¬ 𝑦 ∈ {𝑥})
108		eldif 3914	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑢 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑢 ∧ ¬ 𝑦 ∈ {𝑥}))
109	3, 20	lspssid 21117	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
110	12, 19, 109	syl2an2r 697	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
111	110	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
112	111	sseld 3935	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ (𝑢 ∖ {𝑥}) → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
113	108, 112	biimtrrid 246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → ((𝑦 ∈ 𝑢 ∧ ¬ 𝑦 ∈ {𝑥}) → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
114	107, 113	mpan2d 706	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ 𝑢 → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
115	114	reximdva 3177	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → (∃𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∃𝑢 ∈ 𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
116		eluni2 4875	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑦 ∈ 𝑢)
117		eliun 4959	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
118	115, 116, 117	3imtr4g 299	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → (𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
119	105, 118	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → 𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
120	119	ex 417	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → 𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
121	120	ssrdv 3942	. . 3 ⊢ (𝜑 → (∪ 𝐴 ∖ {𝑥}) ⊆ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
122	121, 9	sseqtrrdi 3977	. 2 ⊢ (𝜑 → (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇)
123	103, 122	jca 520	1 ⊢ (𝜑 → (𝑇 ∈ 𝑃 ∧ (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3415   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  {csn 4588  ∪ cuni 4871  ∪ ciun 4955   Or wor 5567  ‘cfv 6536  (class class class)co 7412   [⊊] crpss 7721  Basecbs 17275  +_gcplusg 17316  Scalarcsca 17319   ·_𝑠 cvsca 17320  LModclmod 20992  LSubSpclss 21063  LSpanclspn 21103  LBasisclbs 21206  LVecclvec 21234

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-rpss 7722  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-er 8692  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-plusg 17329  df-0g 17500  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-grp 19009  df-minusg 19010  df-sbg 19011  df-mgp 20223  df-ur 20270  df-ring 20323  df-lmod 20994  df-lss 21064  df-lsp 21104  df-lvec 21235

This theorem is used by:  lbsextlem3  21295