Theorem lbsextlem2 21464

Description: Lemma for lbsext 21468. 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 2764	. . 3 ⊢ (𝜑 → (Scalar‘𝑊) = (Scalar‘𝑊))
2		eqidd 2764	. . 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 2764	. . 3 ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝑊))
6		eqidd 2764	. . 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 21408	. . . . . . . 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 4010	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ 𝒫 𝑉
16	13, 15	sstrdi 3925	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑉)
17	16	sselda 3913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝒫 𝑉)
18	17	elpwid 4569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → 𝑢 ⊆ 𝑉)
19	18	ssdifssd 4086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ 𝑉)
20		lbsext.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
21	3, 20	lspssv 21285	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
22	12, 19, 21	syl2an2r 698	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
23	22	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
24		iunss 5006	. . . . 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 3951	. . 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 21278	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
30	12, 19, 29	syl2an2r 698	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
31	7	lssn0 21242	. . . . . . . 8 ⊢ ((𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃 → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
32	30, 31	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
33	32	ralrimiva 3158	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
34		r19.2z 4458	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅) → ∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
35	28, 33, 34	syl2anc 596	. . . . 5 ⊢ (𝜑 → ∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
36		iunn0 5028	. . . . 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 3026	. . 3 ⊢ (𝜑 → 𝑇 ≠ ∅)
39	9	eleq2i 2856	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝑇 ↔ 𝑣 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
40		eliun 4958	. . . . . . . . 9 ⊢ (𝑣 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
41		difeq1 4059	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∖ {𝑥}) = (𝑚 ∖ {𝑥}))
42	41	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑚 ∖ {𝑥})))
43	42	eleq2d 2850	. . . . . . . . . 10 ⊢ (𝑢 = 𝑚 → (𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥}))))
44	43	cbvrexvw 3245	. . . . . . . . 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 2856	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑇 ↔ 𝑤 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
47		eliun 4958	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
48		difeq1 4059	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑛 → (𝑢 ∖ {𝑥}) = (𝑛 ∖ {𝑥}))
49	48	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑛 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑛 ∖ {𝑥})))
50	49	eleq2d 2850	. . . . . . . . . 10 ⊢ (𝑢 = 𝑛 → (𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
51	50	cbvrexvw 3245	. . . . . . . . 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 640	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ (∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
54		reeanv 3238	. . . . . . 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 1216	. . . . . . . . . . . 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 1155	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴))
60		sorpssun 7767	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴)) → (𝑚 ∪ 𝑛) ∈ 𝐴)
61	58, 59, 60	syl2anc 596	. . . . . . . . . 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 4902	. . . . . . . . . . . . . . 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 5063	. . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∪ 𝑛) ⊆ 𝑉)
69	68	ssdifssd 4086	. . . . . . . . . . . 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 21278	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ ((𝑚 ∪ 𝑛) ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})) ∈ 𝑃)
71	62, 69, 70	syl2anc 596	. . . . . . . . . . 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 1217	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
73		ssun1 4179	. . . . . . . . . . . . . 14 ⊢ 𝑚 ⊆ (𝑚 ∪ 𝑛)
74		ssdif 4083	. . . . . . . . . . . . . 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 21286	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 1220	. . . . . . . . . . . 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 3914	. . . . . . . . . . 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 4180	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ (𝑚 ∪ 𝑛)
81		ssdif 4083	. . . . . . . . . . . . . 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 21286	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 1221	. . . . . . . . . . . 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 3914	. . . . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
88		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
89		eqid 2763	. . . . . . . . . . . 12 ⊢ (+g‘𝑊) = (+g‘𝑊)
90		eqid 2763	. . . . . . . . . . . 12 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
91	87, 88, 89, 90, 7	lsscl 21244	. . . . . . . . . . 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 1400	. . . . . . . . . 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 4059	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑚 ∪ 𝑛) → (𝑢 ∖ {𝑥}) = ((𝑚 ∪ 𝑛) ∖ {𝑥}))
94	93	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑚 ∪ 𝑛) → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
95	94	eliuni 4960	. . . . . . . . . 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 596	. . . . . . . . 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 2875	. . . . . . . 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 1139	. . . . . . 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 3223	. . . . . 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 436	. . . 4 ⊢ (𝜑 → (𝑟 ∈ (Base‘(Scalar‘𝑊)) → (𝑣 ∈ 𝑇 → (𝑤 ∈ 𝑇 → ((𝑟( ·𝑠 ‘𝑊)𝑣)(+g‘𝑊)𝑤) ∈ 𝑇))))
102	101	3imp2 1368	. . 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 21237	. 2 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
104		eldifi 4070	. . . . . . 7 ⊢ (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → 𝑦 ∈ ∪ 𝐴)
105	104	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → 𝑦 ∈ ∪ 𝐴)
106		eldifn 4071	. . . . . . . . . 10 ⊢ (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → ¬ 𝑦 ∈ {𝑥})
107	106	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → ¬ 𝑦 ∈ {𝑥})
108		eldif 4052	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑢 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑢 ∧ ¬ 𝑦 ∈ {𝑥}))
109	3, 20	lspssid 21287	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
110	12, 19, 109	syl2an2r 698	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
111	110	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
112	111	sseld 3912	. . . . . . . . . 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 707	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ 𝑢 → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
115	114	reximdva 3179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → (∃𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∃𝑢 ∈ 𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
116		eluni2 4874	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑦 ∈ 𝑢)
117		eliun 4958	. . . . . . 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 418	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → 𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
121	120	ssrdv 3919	. . 3 ⊢ (𝜑 → (∪ 𝐴 ∖ {𝑥}) ⊆ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
122	121, 9	sseqtrrdi 3954	. 2 ⊢ (𝜑 → (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇)
123	103, 122	jca 521	1 ⊢ (𝜑 → (𝑇 ∈ 𝑃 ∧ (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560  {csn 4588  ∪ cuni 4870  ∪ ciun 4954   Or wor 5562  ‘cfv 6555  (class class class)co 7440   [⊊] crpss 7759  Basecbs 17413  +_gcplusg 17454  Scalarcsca 17457   ·_𝑠 cvsca 17458  LModclmod 21162  LSubSpclss 21233  LSpanclspn 21273  LBasisclbs 21376  LVecclvec 21404

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-rpss 7760  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-plusg 17467  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-grp 19173  df-minusg 19174  df-sbg 19175  df-mgp 20388  df-ur 20435  df-ring 20488  df-lmod 21164  df-lss 21234  df-lsp 21274  df-lvec 21405

This theorem is used by:  lbsextlem3  21465