Theorem lspfixed 21205

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 2763	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2763	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2763	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
7		lspfixed.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspfixed.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 21180	. . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	8, 9	syl 17	. . . 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 21168	. . 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 234	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))
15	10	3ad2ant1 1147	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LMod)
16		eqid 2763	. . . . . . . . . 10 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
17	2, 16, 7	lspsncl 21051	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
18	10, 12, 17	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19	18	3ad2ant1 1147	. . . . . . 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 1147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LVec)
21	4	lvecdrng 21179	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
22	20, 21	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → (Scalar‘𝑊) ∈ DivRing)
23		simp2l 1214	. . . . . . . 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 1147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26		simpl3 1208	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 7411	. . . . . . . . . . . . . . 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 1206	. . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
33		lspfixed.o	. . . . . . . . . . . . . . . . 17 ⊢ 0 = (0g‘𝑊)
34	2, 4, 6, 32, 33	lmod0vs 20969	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
35	30, 31, 34	syl2anc 593	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 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 7411	. . . . . . . . . . . . 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 1215	. . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . . 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 20952	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
41	15, 38, 39, 40	syl3anc 1392	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 20966	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 + (𝑙( ·𝑠 ‘𝑊)𝑍)) = (𝑙( ·𝑠 ‘𝑊)𝑍))
44	30, 42, 43	syl2anc 593	. . . . . . . . . . . . 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 2802	. . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 1242	. . . . . . . . . . . . 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 21067	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
49	10, 12, 48	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑍}))
50	29, 49	syl 17	. . . . . . . . . . . . 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 21029	. . . . . . . . . . . . 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 849	. . . . . . . . . . . 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 2863	. . . . . . . . . . 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 416	. . . . . . . . . 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 2972	. . . . . . . . 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 15	. . . . . . . 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 2763	. . . . . . . . 9 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
58	5, 32, 57	drnginvrcl 20811	. . . . . . . 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 1392	. . . . . . 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 1147	. . . . . . . 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 849	. . . . . . 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 21029	. . . . . . 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 849	. . . . . 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 20812	. . . . . . . 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 1392	. . . . . . 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 1147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68		simpl3 1208	. . . . . . . . . . . . 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 7403	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = (0g‘(Scalar‘𝑊)) → (𝑙( ·𝑠 ‘𝑊)𝑍) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑍))
70	2, 4, 6, 32, 33	lmod0vs 20969	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑍) = 0 )
71	15, 39, 70	syl2anc 593	. . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . . 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 20952	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
76	15, 23, 74, 75	syl3anc 1392	. . . . . . . . . . . . . . 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 20967	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·𝑠 ‘𝑊)𝑌))
78	15, 76, 77	syl2anc 593	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 2802	. . . . . . . . . . . 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 21051	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
82	10, 11, 81	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83	82	3ad2ant1 1147	. . . . . . . . . . . . . 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 21067	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
85	10, 11, 84	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
86	85	3ad2ant1 1147	. . . . . . . . . . . . . 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 21029	. . . . . . . . . . . . . 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 849	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 2863	. . . . . . . . . . 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 416	. . . . . . . . . 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 2972	. . . . . . . . 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 15	. . . . . . . 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 1206	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 4694	. . . . . . . . . . . . . 14 ⊢ (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
97	96	fveq2d 6871	. . . . . . . . . . . . 13 ⊢ (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
98	2, 33, 7, 15, 74	lsppr0 21166	. . . . . . . . . . . . 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 2820	. . . . . . . . . . . 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 2865	. . . . . . . . . . 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 416	. . . . . . . . . 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 2972	. . . . . . . . 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 15	. . . . . . . 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 21186	. . . . . . . 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 723	. . . . . . 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 21186	. . . . . . 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 723	. . . . . 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 4747	. . . . . 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 592	. . . . 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 1152	. . . . . . 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 20949	. . . . . . . . 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 1392	. . . . . . . 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 21067	. . . . . . . 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 593	. . . . . . 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 2863	. . . . . 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 21191	. . . . . . . 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 1396	. . . . . . 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 20959	. . . . . . . . . . 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 1393	. . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
121		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
122	5, 32, 120, 121, 57	drnginvrl 20814	. . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . 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 7411	. . . . . . . . . . . 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 20961	. . . . . . . . . . . . 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 1393	. . . . . . . . . . . 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 20964	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
128	15, 74, 127	syl2anc 593	. . . . . . . . . . . 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 2806	. . . . . . . . . . 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 7411	. . . . . . . . . 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 2798	. . . . . . . . 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 4595	. . . . . . . 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 6871	. . . . . . 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 2800	. . . . . 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 2865	. . . . 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 7404	. . . . . . . . 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 4595	. . . . . . . 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 6871	. . . . . . 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 2849	. . . . . 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 3582	. . . . 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 593	. . . 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 1133	. . 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 3219	. 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 15	1 ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3087   ∖ cdif 3902  {csn 4583  {cpr 4585  ‘cfv 6521  (class class class)co 7396  Basecbs 17255  +_gcplusg 17296  ._rcmulr 17297  Scalarcsca 17299   ·_𝑠 cvsca 17300  0_gc0g 17478  1_rcur 20241  inv_rcinvr 20446  DivRingcdr 20788  LModclmod 20934  LSubSpclss 21005  LSpanclspn 21045  LVecclvec 21176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-tpos 8206  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-nn 12221  df-2 12290  df-3 12291  df-sets 17210  df-slot 17228  df-ndx 17240  df-base 17256  df-ress 17277  df-plusg 17309  df-mulr 17310  df-0g 17480  df-mgm 18684  df-sgrp 18763  df-mnd 18779  df-submnd 18828  df-grp 18988  df-minusg 18989  df-sbg 18990  df-subg 19175  df-cntz 19367  df-lsm 19686  df-cmn 19832  df-abl 19833  df-mgp 20197  df-rng 20209  df-ur 20242  df-ring 20295  df-oppr 20396  df-dvdsr 20416  df-unit 20417  df-invr 20447  df-drng 20790  df-lmod 20936  df-lss 21006  df-lsp 21046  df-lvec 21177

This theorem is referenced by:  lsatfixedN  39638