Theorem lbsextlem2 20995

Description: Lemma for lbsext 20999. 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 2725	. . 3 ⊢ (𝜑 → (Scalar‘𝑊) = (Scalar‘𝑊))
2		eqidd 2725	. . 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 2725	. . 3 ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝑊))
6		eqidd 2725	. . 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 20939	. . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
12	10, 11	syl 17	. . . . . . 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 4072	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ 𝒫 𝑉
16	13, 15	sstrdi 3986	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑉)
17	16	sselda 3974	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝒫 𝑉)
18	17	elpwid 4603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → 𝑢 ⊆ 𝑉)
19	18	ssdifssd 4134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ 𝑉)
20		lbsext.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
21	3, 20	lspssv 20815	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
22	12, 19, 21	syl2an2r 682	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
23	22	ralrimiva 3138	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
24		iunss 5038	. . . . 5 ⊢ (∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉 ↔ ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
25	23, 24	sylibr 233	. . . 4 ⊢ (𝜑 → ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
26	9, 25	eqsstrid 4022	. . 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 20808	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
30	12, 19, 29	syl2an2r 682	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
31	7	lssn0 20772	. . . . . . . 8 ⊢ ((𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃 → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
33	32	ralrimiva 3138	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
34		r19.2z 4486	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅) → ∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
35	28, 33, 34	syl2anc 583	. . . . 5 ⊢ (𝜑 → ∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
36		iunn0 5060	. . . . 5 ⊢ (∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅ ↔ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
37	35, 36	sylib 217	. . . 4 ⊢ (𝜑 → ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
38	27, 37	eqnetrd 3000	. . 3 ⊢ (𝜑 → 𝑇 ≠ ∅)
39	9	eleq2i 2817	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝑇 ↔ 𝑣 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
40		eliun 4991	. . . . . . . . 9 ⊢ (𝑣 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
41		difeq1 4107	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∖ {𝑥}) = (𝑚 ∖ {𝑥}))
42	41	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑚 ∖ {𝑥})))
43	42	eleq2d 2811	. . . . . . . . . 10 ⊢ (𝑢 = 𝑚 → (𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥}))))
44	43	cbvrexvw 3227	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
45	39, 40, 44	3bitri 297	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑇 ↔ ∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
46	9	eleq2i 2817	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑇 ↔ 𝑤 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
47		eliun 4991	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
48		difeq1 4107	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑛 → (𝑢 ∖ {𝑥}) = (𝑛 ∖ {𝑥}))
49	48	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑛 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑛 ∖ {𝑥})))
50	49	eleq2d 2811	. . . . . . . . . 10 ⊢ (𝑢 = 𝑛 → (𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
51	50	cbvrexvw 3227	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
52	46, 47, 51	3bitri 297	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑇 ↔ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
53	45, 52	anbi12i 626	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ (∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
54		reeanv 3218	. . . . . . 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 278	. . . . . 6 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ ∃𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
56		simp1l 1194	. . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → [⊊] Or 𝐴)
59		simp2 1134	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴))
60		sorpssun 7713	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴)) → (𝑚 ∪ 𝑛) ∈ 𝐴)
61	58, 59, 60	syl2anc 583	. . . . . . . . . 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 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑊 ∈ LMod)
63		elssuni 4931	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∪ 𝑛) ∈ 𝐴 → (𝑚 ∪ 𝑛) ⊆ ∪ 𝐴)
64	61, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∪ 𝑛) ⊆ ∪ 𝐴)
65		sspwuni 5093	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ 𝒫 𝑉 ↔ ∪ 𝐴 ⊆ 𝑉)
66	16, 65	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∪ 𝐴 ⊆ 𝑉)
67	56, 66	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ∪ 𝐴 ⊆ 𝑉)
68	64, 67	sstrd 3984	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∪ 𝑛) ⊆ 𝑉)
69	68	ssdifssd 4134	. . . . . . . . . . . 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 20808	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ ((𝑚 ∪ 𝑛) ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})) ∈ 𝑃)
71	62, 69, 70	syl2anc 583	. . . . . . . . . . 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 1195	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
73		ssun1 4164	. . . . . . . . . . . . . 14 ⊢ 𝑚 ⊆ (𝑚 ∪ 𝑛)
74		ssdif 4131	. . . . . . . . . . . . . 14 ⊢ (𝑚 ⊆ (𝑚 ∪ 𝑛) → (𝑚 ∖ {𝑥}) ⊆ ((𝑚 ∪ 𝑛) ∖ {𝑥}))
75	73, 74	mp1i 13	. . . . . . . . . . . . 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 20816	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 1198	. . . . . . . . . . . 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 3975	. . . . . . . . . . 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 4165	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ (𝑚 ∪ 𝑛)
81		ssdif 4131	. . . . . . . . . . . . . 14 ⊢ (𝑛 ⊆ (𝑚 ∪ 𝑛) → (𝑛 ∖ {𝑥}) ⊆ ((𝑚 ∪ 𝑛) ∖ {𝑥}))
82	80, 81	mp1i 13	. . . . . . . . . . . . 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 20816	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 1199	. . . . . . . . . . . 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 3975	. . . . . . . . . . 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 2724	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
88		eqid 2724	. . . . . . . . . . . 12 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
89		eqid 2724	. . . . . . . . . . . 12 ⊢ (+g‘𝑊) = (+g‘𝑊)
90		eqid 2724	. . . . . . . . . . . 12 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
91	87, 88, 89, 90, 7	lsscl 20774	. . . . . . . . . . 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 1369	. . . . . . . . . 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 4107	. . . . . . . . . . . 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 4993	. . . . . . . . . 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 583	. . . . . . . . 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 2836	. . . . . . . 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 1118	. . . . . . 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 3203	. . . . . 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 241	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) → ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇))
101	100	exp4b 430	. . . 4 ⊢ (𝜑 → (𝑟 ∈ (Base‘(Scalar‘𝑊)) → (𝑣 ∈ 𝑇 → (𝑤 ∈ 𝑇 → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇))))
102	101	3imp2 1346	. . 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 20767	. 2 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
104		eldifi 4118	. . . . . . 7 ⊢ (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → 𝑦 ∈ ∪ 𝐴)
105	104	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → 𝑦 ∈ ∪ 𝐴)
106		eldifn 4119	. . . . . . . . . 10 ⊢ (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → ¬ 𝑦 ∈ {𝑥})
107	106	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → ¬ 𝑦 ∈ {𝑥})
108		eldif 3950	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑢 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑢 ∧ ¬ 𝑦 ∈ {𝑥}))
109	3, 20	lspssid 20817	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
110	12, 19, 109	syl2an2r 682	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
111	110	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
112	111	sseld 3973	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ (𝑢 ∖ {𝑥}) → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
113	108, 112	biimtrrid 242	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → ((𝑦 ∈ 𝑢 ∧ ¬ 𝑦 ∈ {𝑥}) → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
114	107, 113	mpan2d 691	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ 𝑢 → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
115	114	reximdva 3160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → (∃𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∃𝑢 ∈ 𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
116		eluni2 4903	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑦 ∈ 𝑢)
117		eliun 4991	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
118	115, 116, 117	3imtr4g 296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → (𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
119	105, 118	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → 𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
120	119	ex 412	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → 𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
121	120	ssrdv 3980	. . 3 ⊢ (𝜑 → (∪ 𝐴 ∖ {𝑥}) ⊆ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
122	121, 9	sseqtrrdi 4025	. 2 ⊢ (𝜑 → (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇)
123	103, 122	jca 511	1 ⊢ (𝜑 → (𝑇 ∈ 𝑃 ∧ (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  ∃wrex 3062  {crab 3424   ∖ cdif 3937   ∪ cun 3938   ⊆ wss 3940  ∅c0 4314  𝒫 cpw 4594  {csn 4620  ∪ cuni 4899  ∪ ciun 4987   Or wor 5577  ‘cfv 6533  (class class class)co 7401   [⊊] crpss 7705  Basecbs 17140  +_gcplusg 17193  Scalarcsca 17196   ·_𝑠 cvsca 17197  LModclmod 20691  LSubSpclss 20763  LSpanclspn 20803  LBasisclbs 20907  LVecclvec 20935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-rpss 7706  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-0g 17383  df-mgm 18560  df-sgrp 18639  df-mnd 18655  df-grp 18853  df-minusg 18854  df-sbg 18855  df-mgp 20025  df-ur 20072  df-ring 20125  df-lmod 20693  df-lss 20764  df-lsp 20804  df-lvec 20936

This theorem is referenced by:  lbsextlem3  20996