Theorem lspexch 18950

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 18951 vs. lspexchn2 18952); look for lspexch 18950 and prcom 4211 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 2610	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
4		eqid 2610	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2610	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2610	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
7		lspexch.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspexch.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 18927	. . . . 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 18915	. . 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 221	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)))
15		eqid 2610	. . . . . . . 8 ⊢ (-g‘𝑊) = (-g‘𝑊)
16		eqid 2610	. . . . . . . 8 ⊢ (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊))
17	8	3ad2ant1 1075	. . . . . . . . 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 1081	. . . . . . . 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 1075	. . . . . . . . 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 3552	. . . . . . . 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 1075	. . . . . . . 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 18742	. . . . . . 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 1056	. . . . . . . . 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 2616	. . . . . . . 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 1075	. . . . . . . . . 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 18693	. . . . . . . . . 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 18703	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
31	18, 19, 23, 30	syl3anc 1318	. . . . . . . . 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 1080	. . . . . . . . . 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 1075	. . . . . . . . . 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 18703	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑗( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
35	18, 32, 33, 34	syl3anc 1318	. . . . . . . . 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 17326	. . . . . . . . 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 1320	. . . . . . . 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 246	. . . . . . 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 2646	. . . . . 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 2610	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
41		eqid 2610	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
42		lspexch.q	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43	42	3ad2ant1 1075	. . . . . . . . 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 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
49	48	oveq1d 6564	. . . . . . . . . . . . . . 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 18719	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
51	18, 33, 50	syl2anc 691	. . . . . . . . . . . . . . . . 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 6564	. . . . . . . . . . . . . . . 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 18716	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 (+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑘( ·𝑠 ‘𝑊)𝑍))
54	18, 31, 53	syl2anc 691	. . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . 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 2666	. . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . 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 18822	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 2688	. . . . . . . . . . . 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 4261	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 18936	. . . . . . . . . . 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 449	. . . . . . . . . 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 2803	. . . . . . . . 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 4260	. . . . . . . 8 ⊢ (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
69	32, 67, 68	sylanbrc 695	. . . . . . 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 18695	. . . . . . . . . . 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 17290	. . . . . . . . . 10 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
73	71, 19, 72	syl2anc 691	. . . . . . . . 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 18703	. . . . . . . . 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 1318	. . . . . . . 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 18700	. . . . . . . 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 1318	. . . . . . 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 18934	. . . . . 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 221	. . . . 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 2610	. . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
81	2, 80, 7, 18, 22, 23	lspprcl 18799	. . . . . 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 18926	. . . . . . . 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 18587	. . . . . . 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 1318	. . . . . 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 2610	. . . . . . . . . 10 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
87	2, 4, 6, 86	lmodvs1 18714	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑋) = 𝑋)
88	18, 22, 87	syl2anc 691	. . . . . . . 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 6564	. . . . . . 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 18694	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
91	5, 86	ringidcl 18391	. . . . . . . . 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 18911	. . . . . . 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 2689	. . . . . 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 18776	. . . . . 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 1319	. . . . 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 2688	. . . 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 1256	. . 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 3019	. 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923  Grpcgrp 17245  inv_gcminusg 17246  -_gcsg 17247  1_rcur 18324  Ringcrg 18370  inv_rcinvr 18494  DivRingcdr 18570  LModclmod 18686  LSubSpclss 18753  LSpanclspn 18792  LVecclvec 18923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-cntz 17573  df-lsm 17874  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-drng 18572  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lvec 18924

This theorem is referenced by:  lspexchn1  18951  lspindp1  18954  mapdh8ab  36084  mapdh8ad  36086  mapdh8b  36087  mapdh8c  36088  mapdh8e  36091