Theorem lspexch 19636

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 19637 versus lspexchn2 19638); look for lspexch 19636 and prcom 4539 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 2773	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
4		eqid 2773	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2773	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2773	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
7		lspexch.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspexch.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 19613	. . . . 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 19601	. . 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 224	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)))
15		eqid 2773	. . . . . . . 8 ⊢ (-g‘𝑊) = (-g‘𝑊)
16		eqid 2773	. . . . . . . 8 ⊢ (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊))
17	8	3ad2ant1 1114	. . . . . . . . 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 1181	. . . . . . . 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 1114	. . . . . . . . 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 3836	. . . . . . . 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 1114	. . . . . . . 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 19425	. . . . . . 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 1119	. . . . . . . . 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 2779	. . . . . . . 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 1114	. . . . . . . . . 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 19376	. . . . . . . . . 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 19386	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
31	18, 19, 23, 30	syl3anc 1352	. . . . . . . . 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 1180	. . . . . . . . . 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 1114	. . . . . . . . . 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 19386	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑗( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
35	18, 32, 33, 34	syl3anc 1352	. . . . . . . . 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 17987	. . . . . . . . 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 1353	. . . . . . . 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 249	. . . . . . 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 2811	. . . . . 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 2773	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
41		eqid 2773	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
42		lspexch.q	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43	42	3ad2ant1 1114	. . . . . . . . 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 473	. . . . . . . . . . . 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 473	. . . . . . . . . . . 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 473	. . . . . . . . . . . . . 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 6982	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
49	48	oveq1d 6990	. . . . . . . . . . . . . . 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 19402	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
51	18, 33, 50	syl2anc 576	. . . . . . . . . . . . . . . . 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 6990	. . . . . . . . . . . . . . . 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 19399	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 (+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑘( ·𝑠 ‘𝑊)𝑍))
54	18, 31, 53	syl2anc 576	. . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . 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 19508	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 2861	. . . . . . . . . . . 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 4593	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 19622	. . . . . . . . . . 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 405	. . . . . . . . . 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 2983	. . . . . . . . 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 4590	. . . . . . . 8 ⊢ (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
69	32, 67, 68	sylanbrc 575	. . . . . . 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 19378	. . . . . . . . . . 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 17951	. . . . . . . . . 10 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
73	71, 19, 72	syl2anc 576	. . . . . . . . 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 19386	. . . . . . . . 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 1352	. . . . . . . 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 19383	. . . . . . . 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 1352	. . . . . . 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 19620	. . . . . 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 224	. . . . 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 2773	. . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
81	2, 80, 7, 18, 22, 23	lspprcl 19485	. . . . . 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 19612	. . . . . . . 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 19255	. . . . . . 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 1352	. . . . . 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 2773	. . . . . . . . . 10 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
87	2, 4, 6, 86	lmodvs1 19397	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑋) = 𝑋)
88	18, 22, 87	syl2anc 576	. . . . . . . 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 6990	. . . . . . 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 19377	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
91	5, 86	ringidcl 19054	. . . . . . . . 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 19597	. . . . . . 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 2862	. . . . . 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 19462	. . . . . 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 827	. . . . 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 2861	. . . 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 1100	. . 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 3233	. 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 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   ≠ wne 2962  ∃wrex 3084   ∖ cdif 3821  {csn 4436  {cpr 4438  ‘cfv 6186  (class class class)co 6975  Basecbs 16338  +_gcplusg 16420  Scalarcsca 16423   ·_𝑠 cvsca 16424  0_gc0g 16568  Grpcgrp 17904  inv_gcminusg 17905  -_gcsg 17906  1_rcur 18987  Ringcrg 19033  inv_rcinvr 19157  DivRingcdr 19238  LModclmod 19369  LSubSpclss 19438  LSpanclspn 19478  LVecclvec 19609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-tpos 7694  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-nn 11439  df-2 11502  df-3 11503  df-ndx 16341  df-slot 16342  df-base 16344  df-sets 16345  df-ress 16346  df-plusg 16433  df-mulr 16434  df-0g 16570  df-mgm 17723  df-sgrp 17765  df-mnd 17776  df-submnd 17817  df-grp 17907  df-minusg 17908  df-sbg 17909  df-subg 18073  df-cntz 18231  df-lsm 18535  df-cmn 18681  df-abl 18682  df-mgp 18976  df-ur 18988  df-ring 19035  df-oppr 19109  df-dvdsr 19127  df-unit 19128  df-invr 19158  df-drng 19240  df-lmod 19371  df-lss 19439  df-lsp 19479  df-lvec 19610

This theorem is referenced by:  lspexchn1  19637  lspindp1  19640  mapdh8ab  38391  mapdh8ad  38393  mapdh8b  38394  mapdh8c  38395  mapdh8e  38398