Theorem lspfixed 21263

Description: Show membership in the span of the sum of two vectors, one of which (𝑌) is fixed in advance. (Contributed by NM, 27-May-2015.) (Revised by AV, 12-Jul-2022.)

Hypotheses

Ref	Expression
lspfixed.v	⊢ 𝑉 = (Base‘𝑊)
lspfixed.p	⊢ + = (+g‘𝑊)
lspfixed.o	⊢ 0 = (0g‘𝑊)
lspfixed.n	⊢ 𝑁 = (LSpan‘𝑊)
lspfixed.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspfixed.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lspfixed.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
lspfixed.e	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
lspfixed.f	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
lspfixed.g	⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

Assertion

Ref	Expression
lspfixed	⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))

Distinct variable groups:   𝑧,𝑁   𝑧, 0   𝑧, +   𝑧,𝑊   𝑧,𝑋   𝑧,𝑌   𝑧,𝑍

Allowed substitution hints:   𝜑(𝑧)   𝑉(𝑧)

Proof of Theorem lspfixed

Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspfixed.g	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
2		lspfixed.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
3		lspfixed.p	. . . 4 ⊢ + = (+g‘𝑊)
4		eqid 2762	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2762	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2762	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
7		lspfixed.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspfixed.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 21238	. . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
11		lspfixed.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12		lspfixed.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
13	2, 3, 4, 5, 6, 7, 10, 11, 12	lspprel 21226	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))))
14	1, 13	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))
15	10	3ad2ant1 1150	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LMod)
16		eqid 2762	. . . . . . . . . 10 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
17	2, 16, 7	lspsncl 21109	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
18	10, 12, 17	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19	18	3ad2ant1 1150	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
20	8	3ad2ant1 1150	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LVec)
21	4	lvecdrng 21237	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
22	20, 21	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (Scalar‘𝑊) ∈ DivRing)
23		simp2l 1217	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
24		lspfixed.f	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
25	24	3ad2ant1 1150	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26		simpl3 1211	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))
27		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑘 = (0g‘(Scalar‘𝑊)))
28	27	oveq1d 7427	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑘( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
29		simpl1 1209	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝜑)
30	29, 10	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑊 ∈ LMod)
31	29, 11	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑌 ∈ 𝑉)
32		eqid 2762	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
33		lspfixed.o	. . . . . . . . . . . . . . . . 17 ⊢ 0 = (0g‘𝑊)
34	2, 4, 6, 32, 33	lmod0vs 21027	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
35	30, 31, 34	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
36	28, 35	eqtrd 2797	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑘( ·𝑠 ‘𝑊)𝑌) = 0 )
37	36	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) = ( 0 + (𝑙( ·𝑠 ‘𝑊)𝑍)))
38		simp2r 1218	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
39	12	3ad2ant1 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑍 ∈ 𝑉)
40	2, 4, 6, 5	lmodvscl 21010	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
41	15, 38, 39, 40	syl3anc 1397	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
42	41	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
43	2, 3, 33	lmod0vlid 21024	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 + (𝑙( ·𝑠 ‘𝑊)𝑍)) = (𝑙( ·𝑠 ‘𝑊)𝑍))
44	30, 42, 43	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → ( 0 + (𝑙( ·𝑠 ‘𝑊)𝑍)) = (𝑙( ·𝑠 ‘𝑊)𝑍))
45	26, 37, 44	3eqtrd 2801	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑋 = (𝑙( ·𝑠 ‘𝑊)𝑍))
46	29, 18	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
47		simpl2r 1245	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
48	2, 7	lspsnid 21125	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
49	10, 12, 48	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑍}))
50	29, 49	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑍 ∈ (𝑁‘{𝑍}))
51	4, 6, 5, 16	lssvscl 21087	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) ∧ (𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))
52	30, 46, 47, 50, 51	syl22anc 851	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))
53	45, 52	eqeltrd 2862	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑋 ∈ (𝑁‘{𝑍}))
54	53	ex 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑘 = (0g‘(Scalar‘𝑊)) → 𝑋 ∈ (𝑁‘{𝑍})))
55	54	necon3bd 2971	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑍}) → 𝑘 ≠ (0g‘(Scalar‘𝑊))))
56	25, 55	mpd 16	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
57		eqid 2762	. . . . . . . . 9 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
58	5, 32, 57	drnginvrcl 20868	. . . . . . . 8 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
59	22, 23, 56, 58	syl3anc 1397	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
60	49	3ad2ant1 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑍 ∈ (𝑁‘{𝑍}))
61	15, 19, 38, 60, 51	syl22anc 851	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))
62	4, 6, 5, 16	lssvscl 21087	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) ∧ (((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑍}))
63	15, 19, 59, 61, 62	syl22anc 851	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑍}))
64	5, 32, 57	drnginvrn0 20869	. . . . . . . 8 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)))
65	22, 23, 56, 64	syl3anc 1397	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)))
66		lspfixed.e	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
67	66	3ad2ant1 1150	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68		simpl3 1211	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))
69		oveq1 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = (0g‘(Scalar‘𝑊)) → (𝑙( ·𝑠 ‘𝑊)𝑍) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑍))
70	2, 4, 6, 32, 33	lmod0vs 21027	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑍) = 0 )
71	15, 39, 70	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑍) = 0 )
72	69, 71	sylan9eqr 2819	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → (𝑙( ·𝑠 ‘𝑊)𝑍) = 0 )
73	72	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) = ((𝑘( ·𝑠 ‘𝑊)𝑌) + 0 ))
74	11	3ad2ant1 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑌 ∈ 𝑉)
75	2, 4, 6, 5	lmodvscl 21010	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
76	15, 23, 74, 75	syl3anc 1397	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
77	2, 3, 33	lmod0vrid 21025	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·𝑠 ‘𝑊)𝑌))
78	15, 76, 77	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·𝑠 ‘𝑊)𝑌))
79	78	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·𝑠 ‘𝑊)𝑌))
80	68, 73, 79	3eqtrd 2801	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → 𝑋 = (𝑘( ·𝑠 ‘𝑊)𝑌))
81	2, 16, 7	lspsncl 21109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
82	10, 11, 81	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83	82	3ad2ant1 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
84	2, 7	lspsnid 21125	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
85	10, 11, 84	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
86	85	3ad2ant1 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑌}))
87	4, 6, 5, 16	lssvscl 21087	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ (𝑁‘{𝑌}))) → (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘{𝑌}))
88	15, 83, 23, 86, 87	syl22anc 851	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘{𝑌}))
89	88	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘{𝑌}))
90	80, 89	eqeltrd 2862	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → 𝑋 ∈ (𝑁‘{𝑌}))
91	90	ex 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑙 = (0g‘(Scalar‘𝑊)) → 𝑋 ∈ (𝑁‘{𝑌})))
92	91	necon3bd 2971	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑌}) → 𝑙 ≠ (0g‘(Scalar‘𝑊))))
93	67, 92	mpd 16	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑙 ≠ (0g‘(Scalar‘𝑊)))
94		simpl1 1209	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝜑)
95	94, 1	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
96		preq2 4699	. . . . . . . . . . . . . 14 ⊢ (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
97	96	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
98	2, 33, 7, 15, 74	lsppr0 21224	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑁‘{𝑌, 0 }) = (𝑁‘{𝑌}))
99	97, 98	sylan9eqr 2819	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑍 = 0 ) → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌}))
100	95, 99	eleqtrd 2864	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑁‘{𝑌}))
101	100	ex 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑍 = 0 → 𝑋 ∈ (𝑁‘{𝑌})))
102	101	necon3bd 2971	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑌}) → 𝑍 ≠ 0 ))
103	67, 102	mpd 16	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑍 ≠ 0 )
104	2, 6, 4, 5, 32, 33, 20, 38, 39	lvecvsn0 21244	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((𝑙( ·𝑠 ‘𝑊)𝑍) ≠ 0 ↔ (𝑙 ≠ (0g‘(Scalar‘𝑊)) ∧ 𝑍 ≠ 0 )))
105	93, 103, 104	mpbir2and 725	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑙( ·𝑠 ‘𝑊)𝑍) ≠ 0 )
106	2, 6, 4, 5, 32, 33, 20, 59, 41	lvecvsn0 21244	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ≠ 0 ↔ (((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)) ∧ (𝑙( ·𝑠 ‘𝑊)𝑍) ≠ 0 )))
107	65, 105, 106	mpbir2and 725	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ≠ 0 )
108		eldifsn 4752	. . . . . 6 ⊢ ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }) ↔ ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑍}) ∧ (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ≠ 0 ))
109	63, 107, 108	sylanbrc 594	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }))
110		simp3 1155	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))
111	2, 3	lmodvacl 21007	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ 𝑉)
112	15, 76, 41, 111	syl3anc 1397	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ 𝑉)
113	2, 7	lspsnid 21125	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ 𝑉) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))}))
114	15, 112, 113	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))}))
115	110, 114	eqeltrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑋 ∈ (𝑁‘{((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))}))
116	2, 4, 6, 5, 32, 7	lspsnvs 21249	. . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ (((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊))) ∧ ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ 𝑉) → (𝑁‘{(((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))}) = (𝑁‘{((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))}))
117	20, 59, 65, 112, 116	syl121anc 1401	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑁‘{(((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))}) = (𝑁‘{((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))}))
118	2, 3, 4, 6, 5	lmodvsdi 21017	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) = ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑌)) + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍))))
119	15, 59, 76, 41, 118	syl13anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) = ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑌)) + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍))))
120		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
121		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
122	5, 32, 120, 121, 57	drnginvrl 20871	. . . . . . . . . . . . . 14 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → (((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘) = (1r‘(Scalar‘𝑊)))
123	22, 23, 56, 122	syl3anc 1397	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘) = (1r‘(Scalar‘𝑊)))
124	123	oveq1d 7427	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘)( ·𝑠 ‘𝑊)𝑌) = ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
125	2, 4, 6, 5, 120	lmodvsass 21019	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉)) → ((((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑌)))
126	15, 59, 23, 74, 125	syl13anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑌)))
127	2, 4, 6, 121	lmodvs1 21022	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
128	15, 74, 127	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
129	124, 126, 128	3eqtr3d 2805	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑌)) = 𝑌)
130	129	oveq1d 7427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑌)) + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍))) = (𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍))))
131	119, 130	eqtrd 2797	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) = (𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍))))
132	131	sneqd 4600	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → {(((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))} = {(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)))})
133	132	fveq2d 6885	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑁‘{(((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)))}))
134	117, 133	eqtr3d 2799	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (𝑁‘{((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)))}))
135	115, 134	eleqtrd 2864	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑋 ∈ (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)))}))
136		oveq2 7420	. . . . . . . . 9 ⊢ (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) → (𝑌 + 𝑧) = (𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍))))
137	136	sneqd 4600	. . . . . . . 8 ⊢ (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) → {(𝑌 + 𝑧)} = {(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)))})
138	137	fveq2d 6885	. . . . . . 7 ⊢ (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) → (𝑁‘{(𝑌 + 𝑧)}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)))}))
139	138	eleq2d 2848	. . . . . 6 ⊢ (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) → (𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}) ↔ 𝑋 ∈ (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)))})))
140	139	rspcev 3580	. . . . 5 ⊢ (((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }) ∧ 𝑋 ∈ (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)(𝑙( ·𝑠 ‘𝑊)𝑍)))})) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
141	109, 135, 140	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
142	141	3exp 1136	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) → (𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))))
143	142	rexlimdvv 3220	. 2 ⊢ (𝜑 → (∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)})))
144	14, 143	mpd 16	1 ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088   ∖ cdif 3901  {csn 4588  {cpr 4590  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316  ._rcmulr 17317  Scalarcsca 17319   ·_𝑠 cvsca 17320  0_gc0g 17498  1_rcur 20269  inv_rcinvr 20476  DivRingcdr 20838  LModclmod 20992  LSubSpclss 21063  LSpanclspn 21103  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-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8220  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-3 12310  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-plusg 17329  df-mulr 17330  df-0g 17500  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-submnd 18848  df-grp 19009  df-minusg 19010  df-sbg 19011  df-subg 19195  df-cntz 19393  df-lsm 19712  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-oppr 20426  df-dvdsr 20446  df-unit 20447  df-invr 20477  df-drng 20840  df-lmod 20994  df-lss 21064  df-lsp 21104  df-lvec 21235

This theorem is used by:  lsatfixedN  39811