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Theorem lbsextlem2 20771

Description: Lemma for lbsext 20775. 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 2733	. . 3 ⊢ (𝜑 → (Scalar‘𝑊) = (Scalar‘𝑊))
2		eqidd 2733	. . 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 2733	. . 3 ⊢ (𝜑 → (+_g‘𝑊) = (+_g‘𝑊))
6		eqidd 2733	. . 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 20716	. . . . . . . 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 4080	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ 𝒫 𝑉
16	13, 15	sstrdi 3994	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑉)
17	16	sselda 3982	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝒫 𝑉)
18	17	elpwid 4611	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → 𝑢 ⊆ 𝑉)
19	18	ssdifssd 4142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ 𝑉)
20		lbsext.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
21	3, 20	lspssv 20593	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
22	12, 19, 21	syl2an2r 683	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
23	22	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
24		iunss 5048	. . . . 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 4030	. . 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 20586	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
30	12, 19, 29	syl2an2r 683	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
31	7	lssn0 20550	. . . . . . . 8 ⊢ ((𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃 → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
33	32	ralrimiva 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
34		r19.2z 4494	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅) → ∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
35	28, 33, 34	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∃𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
36		iunn0 5070	. . . . 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 3008	. . 3 ⊢ (𝜑 → 𝑇 ≠ ∅)
39	9	eleq2i 2825	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝑇 ↔ 𝑣 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
40		eliun 5001	. . . . . . . . 9 ⊢ (𝑣 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
41		difeq1 4115	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∖ {𝑥}) = (𝑚 ∖ {𝑥}))
42	41	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑚 ∖ {𝑥})))
43	42	eleq2d 2819	. . . . . . . . . 10 ⊢ (𝑢 = 𝑚 → (𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥}))))
44	43	cbvrexvw 3235	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
45	39, 40, 44	3bitri 296	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑇 ↔ ∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
46	9	eleq2i 2825	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑇 ↔ 𝑤 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
47		eliun 5001	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
48		difeq1 4115	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑛 → (𝑢 ∖ {𝑥}) = (𝑛 ∖ {𝑥}))
49	48	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑛 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑛 ∖ {𝑥})))
50	49	eleq2d 2819	. . . . . . . . . 10 ⊢ (𝑢 = 𝑛 → (𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
51	50	cbvrexvw 3235	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
52	46, 47, 51	3bitri 296	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑇 ↔ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
53	45, 52	anbi12i 627	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ (∃𝑚 ∈ 𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛 ∈ 𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
54		reeanv 3226	. . . . . . 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 277	. . . . . 6 ⊢ ((𝑣 ∈ 𝑇 ∧ 𝑤 ∈ 𝑇) ↔ ∃𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
56		simp1l 1197	. . . . . . . . . . . 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 1137	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴))
60		sorpssun 7719	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴)) → (𝑚 ∪ 𝑛) ∈ 𝐴)
61	58, 59, 60	syl2anc 584	. . . . . . . . . 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 4941	. . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . 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 3992	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∪ 𝑛) ⊆ 𝑉)
69	68	ssdifssd 4142	. . . . . . . . . . . 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 20586	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ ((𝑚 ∪ 𝑛) ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})) ∈ 𝑃)
71	62, 69, 70	syl2anc 584	. . . . . . . . . . 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 1198	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
73		ssun1 4172	. . . . . . . . . . . . . 14 ⊢ 𝑚 ⊆ (𝑚 ∪ 𝑛)
74		ssdif 4139	. . . . . . . . . . . . . 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 20594	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 1201	. . . . . . . . . . . 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 3983	. . . . . . . . . . 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 4173	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ (𝑚 ∪ 𝑛)
81		ssdif 4139	. . . . . . . . . . . . . 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 20594	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 1202	. . . . . . . . . . . 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 3983	. . . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
88		eqid 2732	. . . . . . . . . . . 12 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
89		eqid 2732	. . . . . . . . . . . 12 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
90		eqid 2732	. . . . . . . . . . . 12 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
91	87, 88, 89, 90, 7	lsscl 20552	. . . . . . . . . . 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 1372	. . . . . . . . . 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 4115	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑚 ∪ 𝑛) → (𝑢 ∖ {𝑥}) = ((𝑚 ∪ 𝑛) ∖ {𝑥}))
94	93	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑚 ∪ 𝑛) → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘((𝑚 ∪ 𝑛) ∖ {𝑥})))
95	94	eliuni 5003	. . . . . . . . . 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 584	. . . . . . . . 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 2844	. . . . . . . 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 1121	. . . . . . 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 3211	. . . . . 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 431	. . . 4 ⊢ (𝜑 → (𝑟 ∈ (Base‘(Scalar‘𝑊)) → (𝑣 ∈ 𝑇 → (𝑤 ∈ 𝑇 → ((𝑟( ·_𝑠 ‘𝑊)𝑣)(+_g‘𝑊)𝑤) ∈ 𝑇))))
102	101	3imp2 1349	. . 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 20545	. 2 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
104		eldifi 4126	. . . . . . 7 ⊢ (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → 𝑦 ∈ ∪ 𝐴)
105	104	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → 𝑦 ∈ ∪ 𝐴)
106		eldifn 4127	. . . . . . . . . 10 ⊢ (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → ¬ 𝑦 ∈ {𝑥})
107	106	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → ¬ 𝑦 ∈ {𝑥})
108		eldif 3958	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑢 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑢 ∧ ¬ 𝑦 ∈ {𝑥}))
109	3, 20	lspssid 20595	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
110	12, 19, 109	syl2an2r 683	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
111	110	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
112	111	sseld 3981	. . . . . . . . . 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 692	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ 𝑢 → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
115	114	reximdva 3168	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐴 ∖ {𝑥})) → (∃𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∃𝑢 ∈ 𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
116		eluni2 4912	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑦 ∈ 𝑢)
117		eliun 5001	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢 ∈ 𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
118	115, 116, 117	3imtr4g 295	. . . . . 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 413	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (∪ 𝐴 ∖ {𝑥}) → 𝑦 ∈ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
121	120	ssrdv 3988	. . 3 ⊢ (𝜑 → (∪ 𝐴 ∖ {𝑥}) ⊆ ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
122	121, 9	sseqtrrdi 4033	. 2 ⊢ (𝜑 → (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇)
123	103, 122	jca 512	1 ⊢ (𝜑 → (𝑇 ∈ 𝑃 ∧ (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 ∪ ciun 4997 Or wor 5587 ‘cfv 6543 (class class class)co 7408 [⊊] crpss 7711 Basecbs 17143 +_gcplusg 17196 Scalarcsca 17199 ·_𝑠 cvsca 17200 LModclmod 20470 LSubSpclss 20541 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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-rpss 7712 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mgp 19987 df-ur 20004 df-ring 20057 df-lmod 20472 df-lss 20542 df-lsp 20582 df-lvec 20713

This theorem is referenced by: lbsextlem3 20772

W3C validator