Theorem lspfixed 21088

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 2737	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2737	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2737	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
7		lspfixed.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspfixed.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 21063	. . . . 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 21051	. . 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 232	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍)))
15	10	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LMod)
16		eqid 2737	. . . . . . . . . 10 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
17	2, 16, 7	lspsncl 20933	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
18	10, 12, 17	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19	18	3ad2ant1 1134	. . . . . . 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 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LVec)
21	4	lvecdrng 21062	. . . . . . . . 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 1201	. . . . . . . 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 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26		simpl3 1195	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
33		lspfixed.o	. . . . . . . . . . . . . . . . 17 ⊢ 0 = (0g‘𝑊)
34	2, 4, 6, 32, 33	lmod0vs 20851	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
35	30, 31, 34	syl2anc 585	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . 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 20834	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
41	15, 38, 39, 40	syl3anc 1374	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 20848	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝑙( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 + (𝑙( ·𝑠 ‘𝑊)𝑍)) = (𝑙( ·𝑠 ‘𝑊)𝑍))
44	30, 42, 43	syl2anc 585	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 1229	. . . . . . . . . . . . 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 20949	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
49	10, 12, 48	syl2anc 585	. . . . . . . . . . . . . 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 20911	. . . . . . . . . . . . 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 839	. . . . . . . . . . . 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 2837	. . . . . . . . . . 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 412	. . . . . . . . . 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 2947	. . . . . . . . 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 2737	. . . . . . . . 9 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
58	5, 32, 57	drnginvrcl 20691	. . . . . . . 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 1374	. . . . . . 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 1134	. . . . . . . 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 839	. . . . . . 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 20911	. . . . . . 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 839	. . . . . 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 20692	. . . . . . . 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 1374	. . . . . . 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 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠 ‘𝑊)𝑌) + (𝑙( ·𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68		simpl3 1195	. . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = (0g‘(Scalar‘𝑊)) → (𝑙( ·𝑠 ‘𝑊)𝑍) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑍))
70	2, 4, 6, 32, 33	lmod0vs 20851	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑍) = 0 )
71	15, 39, 70	syl2anc 585	. . . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . 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 20834	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
76	15, 23, 74, 75	syl3anc 1374	. . . . . . . . . . . . . . 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 20849	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉) → ((𝑘( ·𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·𝑠 ‘𝑊)𝑌))
78	15, 76, 77	syl2anc 585	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 20933	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
82	10, 11, 81	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83	82	3ad2ant1 1134	. . . . . . . . . . . . . 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 20949	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
85	10, 11, 84	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
86	85	3ad2ant1 1134	. . . . . . . . . . . . . 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 20911	. . . . . . . . . . . . . 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 839	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 2837	. . . . . . . . . . 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 412	. . . . . . . . . 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 2947	. . . . . . . . 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 1193	. . . . . . . . . . . . 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 4692	. . . . . . . . . . . . . 14 ⊢ (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
97	96	fveq2d 6839	. . . . . . . . . . . . 13 ⊢ (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
98	2, 33, 7, 15, 74	lsppr0 21049	. . . . . . . . . . . . 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 2794	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 412	. . . . . . . . . 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 2947	. . . . . . . . 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 21069	. . . . . . . 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 714	. . . . . . 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 21069	. . . . . . 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 714	. . . . . 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 4743	. . . . . 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 584	. . . . 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 1139	. . . . . . 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 20831	. . . . . . . . 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 1374	. . . . . . . 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 20949	. . . . . . . 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 585	. . . . . . 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 2837	. . . . . 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 21074	. . . . . . . 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 1378	. . . . . . 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 20841	. . . . . . . . . . 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 1375	. . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
121		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
122	5, 32, 120, 121, 57	drnginvrl 20694	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 7376	. . . . . . . . . . . 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 20843	. . . . . . . . . . . . 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 1375	. . . . . . . . . . . 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 20846	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
128	15, 74, 127	syl2anc 585	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 7376	. . . . . . . . . 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 2772	. . . . . . . . 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 4593	. . . . . . . 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 6839	. . . . . . 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 2774	. . . . . 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 2839	. . . . 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 7369	. . . . . . . . 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 4593	. . . . . . . 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 6839	. . . . . . 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 2823	. . . . . 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 3577	. . . . 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 585	. . . 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 1120	. . 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 3193	. 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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3061   ∖ cdif 3899  {csn 4581  {cpr 4583  ‘cfv 6493  (class class class)co 7361  Basecbs 17141  +_gcplusg 17182  ._rcmulr 17183  Scalarcsca 17185   ·_𝑠 cvsca 17186  0_gc0g 17364  1_rcur 20121  inv_rcinvr 20328  DivRingcdr 20667  LModclmod 20816  LSubSpclss 20887  LSpanclspn 20927  LVecclvec 21059

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-cnex 11087  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107  ax-pre-mulgt0 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-tpos 8171  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12151  df-2 12213  df-3 12214  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17142  df-ress 17163  df-plusg 17195  df-mulr 17196  df-0g 17366  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-submnd 18714  df-grp 18871  df-minusg 18872  df-sbg 18873  df-subg 19058  df-cntz 19251  df-lsm 19570  df-cmn 19716  df-abl 19717  df-mgp 20081  df-rng 20093  df-ur 20122  df-ring 20175  df-oppr 20278  df-dvdsr 20298  df-unit 20299  df-invr 20329  df-drng 20669  df-lmod 20818  df-lss 20888  df-lsp 20928  df-lvec 21060

This theorem is referenced by:  lsatfixedN  39348