Theorem lspexch 20391

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 20392 versus lspexchn2 20393); look for lspexch 20391 and prcom 4668 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 2738	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
4		eqid 2738	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2738	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
7		lspexch.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspexch.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 20368	. . . . 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 20356	. . 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 231	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)))
15		eqid 2738	. . . . . . . 8 ⊢ (-g‘𝑊) = (-g‘𝑊)
16		eqid 2738	. . . . . . . 8 ⊢ (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊))
17	8	3ad2ant1 1132	. . . . . . . . 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 1199	. . . . . . . 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 1132	. . . . . . . . 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 3899	. . . . . . . 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 1132	. . . . . . . 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 20179	. . . . . . 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 1137	. . . . . . . . 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 2744	. . . . . . . 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 1132	. . . . . . . . . 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 20130	. . . . . . . . . 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 20140	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
31	18, 19, 23, 30	syl3anc 1370	. . . . . . . . 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 1198	. . . . . . . . . 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 1132	. . . . . . . . . 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 20140	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑗( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
35	18, 32, 33, 34	syl3anc 1370	. . . . . . . . 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 18663	. . . . . . . . 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 1371	. . . . . . . 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 256	. . . . . . 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 2780	. . . . . 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 2738	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
41		eqid 2738	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
42		lspexch.q	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43	42	3ad2ant1 1132	. . . . . . . . 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 481	. . . . . . . . . . . 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 481	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 7282	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
49	48	oveq1d 7290	. . . . . . . . . . . . . . 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 20156	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
51	18, 33, 50	syl2anc 584	. . . . . . . . . . . . . . . . 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 7290	. . . . . . . . . . . . . . . 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 20153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 (+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑘( ·𝑠 ‘𝑊)𝑍))
54	18, 31, 53	syl2anc 584	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 20263	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 4723	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 20377	. . . . . . . . . . 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 413	. . . . . . . . . 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 2964	. . . . . . . . 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 4720	. . . . . . . 8 ⊢ (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
69	32, 67, 68	sylanbrc 583	. . . . . . 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 20132	. . . . . . . . . . 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 18627	. . . . . . . . . 10 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
73	71, 19, 72	syl2anc 584	. . . . . . . . 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 20140	. . . . . . . . 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 1370	. . . . . . . 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 20137	. . . . . . . 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 1370	. . . . . . 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 20375	. . . . . 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 231	. . . . 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 2738	. . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
81	2, 80, 7, 18, 22, 23	lspprcl 20240	. . . . . 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 20367	. . . . . . . 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 20008	. . . . . . 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 1370	. . . . . 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 2738	. . . . . . . . . 10 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
87	2, 4, 6, 86	lmodvs1 20151	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑋) = 𝑋)
88	18, 22, 87	syl2anc 584	. . . . . . . 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 7290	. . . . . . 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 20131	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
91	5, 86	ringidcl 19807	. . . . . . . . 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 20352	. . . . . . 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 2840	. . . . . 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 20217	. . . . . 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 836	. . . . 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 2839	. . . 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 1118	. . 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 3222	. 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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065   ∖ cdif 3884  {csn 4561  {cpr 4563  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Scalarcsca 16965   ·_𝑠 cvsca 16966  0_gc0g 17150  Grpcgrp 18577  inv_gcminusg 18578  -_gcsg 18579  1_rcur 19737  Ringcrg 19783  inv_rcinvr 19913  DivRingcdr 19991  LModclmod 20123  LSubSpclss 20193  LSpanclspn 20233  LVecclvec 20364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-cntz 18923  df-lsm 19241  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-drng 19993  df-lmod 20125  df-lss 20194  df-lsp 20234  df-lvec 20365

This theorem is referenced by:  lspexchn1  20392  lspindp1  20395  mapdh8ab  39791  mapdh8ad  39793  mapdh8b  39794  mapdh8c  39795  mapdh8e  39798