Theorem lspexch 20120

Description: Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 20121 versus lspexchn2 20122); look for lspexch 20120 and prcom 4634 in same proof. TODO: would a hypothesis of ¬ 𝑋 ∈ (𝑁‘{𝑍}) instead of (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) be better overall? This would be shorter and also satisfy the 𝑋 ≠ 0 condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the ≠ pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)

Hypotheses

Ref	Expression
lspexch.v	⊢ 𝑉 = (Base‘𝑊)
lspexch.o	⊢ 0 = (0g‘𝑊)
lspexch.n	⊢ 𝑁 = (LSpan‘𝑊)
lspexch.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspexch.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lspexch.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lspexch.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
lspexch.q	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
lspexch.e	⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

Assertion

Ref	Expression
lspexch	⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))

Proof of Theorem lspexch

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

Step	Hyp	Ref	Expression
1		lspexch.e	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
2		lspexch.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
3		eqid 2736	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
4		eqid 2736	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2736	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2736	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
7		lspexch.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspexch.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 20097	. . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
11		lspexch.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12		lspexch.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
13	2, 3, 4, 5, 6, 7, 10, 11, 12	lspprel 20085	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))))
14	1, 13	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)))
15		eqid 2736	. . . . . . . 8 ⊢ (-g‘𝑊) = (-g‘𝑊)
16		eqid 2736	. . . . . . . 8 ⊢ (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊))
17	8	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LVec)
18	17, 9	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LMod)
19		simp2r 1202	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
20		lspexch.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
22	21	eldifad 3865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑋 ∈ 𝑉)
23	12	3ad2ant1 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑍 ∈ 𝑉)
24	2, 3, 15, 6, 4, 5, 16, 18, 19, 22, 23	lmodsubvs 19909	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑋(-g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)))
25		simp3 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)))
26	25	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = 𝑋)
27	10	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LMod)
28		lmodgrp 19860	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
29	27, 28	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ Grp)
30	2, 4, 6, 5	lmodvscl 19870	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
31	18, 19, 23, 30	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
32		simp2l 1201	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑗 ∈ (Base‘(Scalar‘𝑊)))
33	11	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑌 ∈ 𝑉)
34	2, 4, 6, 5	lmodvscl 19870	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑗( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
35	18, 32, 33, 34	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑗( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
36	2, 3, 15	grpsubadd 18405	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ (𝑋 ∈ 𝑉 ∧ (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉 ∧ (𝑗( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)) → ((𝑋(-g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑗( ·𝑠 ‘𝑊)𝑌) ↔ ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = 𝑋))
37	29, 22, 31, 35, 36	syl13anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ((𝑋(-g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑗( ·𝑠 ‘𝑊)𝑌) ↔ ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = 𝑋))
38	26, 37	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑋(-g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑗( ·𝑠 ‘𝑊)𝑌))
39	24, 38	eqtr3d 2773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)) = (𝑗( ·𝑠 ‘𝑊)𝑌))
40		eqid 2736	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
41		eqid 2736	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
42		lspexch.q	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43	42	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
44		lspexch.o	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑊)
45	17	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑊 ∈ LVec)
46	23	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑍 ∈ 𝑉)
47	25	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)))
48		oveq1 7198	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
49	48	oveq1d 7206	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (0g‘(Scalar‘𝑊)) → ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)))
50	2, 4, 6, 40, 44	lmod0vs 19886	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
51	18, 33, 50	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
52	51	oveq1d 7206	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = ( 0 (+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)))
53	2, 3, 44	lmod0vlid 19883	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 (+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑘( ·𝑠 ‘𝑊)𝑍))
54	18, 31, 53	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ( 0 (+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑘( ·𝑠 ‘𝑊)𝑍))
55	52, 54	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑘( ·𝑠 ‘𝑊)𝑍))
56	49, 55	sylan9eqr 2793	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑘( ·𝑠 ‘𝑊)𝑍))
57	47, 56	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑋 = (𝑘( ·𝑠 ‘𝑊)𝑍))
58	2, 6, 4, 5, 7, 18, 19, 23	lspsneli 19992	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))
59	58	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))
60	57, 59	eqeltrd 2831	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑋 ∈ (𝑁‘{𝑍}))
61		eldifsni 4689	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
62	21, 61	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑋 ≠ 0 )
63	62	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑋 ≠ 0 )
64	2, 44, 7, 45, 46, 60, 63	lspsneleq 20106	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → (𝑁‘{𝑋}) = (𝑁‘{𝑍}))
65	64	ex 416	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑁‘{𝑋}) = (𝑁‘{𝑍})))
66	65	necon3d 2953	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) → 𝑗 ≠ (0g‘(Scalar‘𝑊))))
67	43, 66	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑗 ≠ (0g‘(Scalar‘𝑊)))
68		eldifsn 4686	. . . . . . . 8 ⊢ (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
69	32, 67, 68	sylanbrc 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
70	4	lmodfgrp 19862	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Grp)
71	27, 70	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (Scalar‘𝑊) ∈ Grp)
72	5, 16	grpinvcl 18369	. . . . . . . . . 10 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
73	71, 19, 72	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
74	2, 4, 6, 5	lmodvscl 19870	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
75	18, 73, 23, 74	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
76	2, 3	lmodvacl 19867	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → (𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)) ∈ 𝑉)
77	18, 22, 75, 76	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)) ∈ 𝑉)
78	2, 6, 4, 5, 40, 41, 17, 69, 77, 33	lvecinv 20104	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ((𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)) = (𝑗( ·𝑠 ‘𝑊)𝑌) ↔ 𝑌 = (((invr‘(Scalar‘𝑊))‘𝑗)( ·𝑠 ‘𝑊)(𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)))))
79	39, 78	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑌 = (((invr‘(Scalar‘𝑊))‘𝑗)( ·𝑠 ‘𝑊)(𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍))))
80		eqid 2736	. . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
81	2, 80, 7, 18, 22, 23	lspprcl 19969	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑁‘{𝑋, 𝑍}) ∈ (LSubSp‘𝑊))
82	4	lvecdrng 20096	. . . . . . . 8 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
83	17, 82	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (Scalar‘𝑊) ∈ DivRing)
84	5, 40, 41	drnginvrcl 19738	. . . . . . 7 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)))
85	83, 32, 67, 84	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)))
86		eqid 2736	. . . . . . . . . 10 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
87	2, 4, 6, 86	lmodvs1 19881	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑋) = 𝑋)
88	18, 22, 87	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑋) = 𝑋)
89	88	oveq1d 7206	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑋)(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)) = (𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)))
90	4	lmodring 19861	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
91	5, 86	ringidcl 19540	. . . . . . . . 9 ⊢ ((Scalar‘𝑊) ∈ Ring → (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
92	18, 90, 91	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
93	2, 3, 6, 4, 5, 7, 18, 92, 73, 22, 23	lsppreli 20081	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑋)(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑋, 𝑍}))
94	89, 93	eqeltrrd 2832	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑋, 𝑍}))
95	4, 6, 5, 80	lssvscl 19946	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑋, 𝑍}) ∈ (LSubSp‘𝑊)) ∧ (((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑋, 𝑍}))) → (((invr‘(Scalar‘𝑊))‘𝑗)( ·𝑠 ‘𝑊)(𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍))) ∈ (𝑁‘{𝑋, 𝑍}))
96	18, 81, 85, 94, 95	syl22anc 839	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑗)( ·𝑠 ‘𝑊)(𝑋(+g‘𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠 ‘𝑊)𝑍))) ∈ (𝑁‘{𝑋, 𝑍}))
97	79, 96	eqeltrd 2831	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))
98	97	3exp 1121	. . 3 ⊢ (𝜑 → ((𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))))
99	98	rexlimdvv 3202	. 2 ⊢ (𝜑 → (∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
100	14, 99	mpd 15	1 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ∃wrex 3052   ∖ cdif 3850  {csn 4527  {cpr 4529  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  +_gcplusg 16749  Scalarcsca 16752   ·_𝑠 cvsca 16753  0_gc0g 16898  Grpcgrp 18319  inv_gcminusg 18320  -_gcsg 18321  1_rcur 19470  Ringcrg 19516  inv_rcinvr 19643  DivRingcdr 19721  LModclmod 19853  LSubSpclss 19922  LSpanclspn 19962  LVecclvec 20093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-tpos 7946  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-2 11858  df-3 11859  df-ndx 16669  df-slot 16670  df-base 16672  df-sets 16673  df-ress 16674  df-plusg 16762  df-mulr 16763  df-0g 16900  df-mgm 18068  df-sgrp 18117  df-mnd 18128  df-submnd 18173  df-grp 18322  df-minusg 18323  df-sbg 18324  df-subg 18494  df-cntz 18665  df-lsm 18979  df-cmn 19126  df-abl 19127  df-mgp 19459  df-ur 19471  df-ring 19518  df-oppr 19595  df-dvdsr 19613  df-unit 19614  df-invr 19644  df-drng 19723  df-lmod 19855  df-lss 19923  df-lsp 19963  df-lvec 20094

This theorem is referenced by:  lspexchn1  20121  lspindp1  20124  mapdh8ab  39477  mapdh8ad  39479  mapdh8b  39480  mapdh8c  39481  mapdh8e  39484