Theorem lspexch 20306

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 20307 versus lspexchn2 20308); look for lspexch 20306 and prcom 4665 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 20283	. . . . 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 20271	. . 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 1131	. . . . . . . . 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 1198	. . . . . . . 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 1131	. . . . . . . . 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 3895	. . . . . . . 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 1131	. . . . . . . 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 20094	. . . . . . 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 1136	. . . . . . . . 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 1131	. . . . . . . . . 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 20045	. . . . . . . . . 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 20055	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉)
31	18, 19, 23, 30	syl3anc 1369	. . . . . . . . 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 1197	. . . . . . . . . 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 1131	. . . . . . . . . 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 20055	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑗( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
35	18, 32, 33, 34	syl3anc 1369	. . . . . . . . 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 18578	. . . . . . . . 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 1370	. . . . . . . 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 1131	. . . . . . . . 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 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
49	48	oveq1d 7270	. . . . . . . . . . . . . . 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 20071	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 0 )
51	18, 33, 50	syl2anc 583	. . . . . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . . . . 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 20068	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 (+g‘𝑊)(𝑘( ·𝑠 ‘𝑊)𝑍)) = (𝑘( ·𝑠 ‘𝑊)𝑍))
54	18, 31, 53	syl2anc 583	. . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . 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 20178	. . . . . . . . . . . . . 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 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 4720	. . . . . . . . . . . . . 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 20292	. . . . . . . . . . 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 412	. . . . . . . . . 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 2963	. . . . . . . . 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 4717	. . . . . . . 8 ⊢ (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
69	32, 67, 68	sylanbrc 582	. . . . . . 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 20047	. . . . . . . . . . 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 18542	. . . . . . . . . 10 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
73	71, 19, 72	syl2anc 583	. . . . . . . . 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 20055	. . . . . . . . 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 1369	. . . . . . . 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 20052	. . . . . . . 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 1369	. . . . . . 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 20290	. . . . . 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 20155	. . . . . 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 20282	. . . . . . . 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 19923	. . . . . . 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 1369	. . . . . 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 20066	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑋) = 𝑋)
88	18, 22, 87	syl2anc 583	. . . . . . . 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 7270	. . . . . . 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 20046	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
91	5, 86	ringidcl 19722	. . . . . . . . 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 20267	. . . . . . 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 20132	. . . . . 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 835	. . . . 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 1117	. . 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 3221	. 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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ∖ cdif 3880  {csn 4558  {cpr 4560  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067  Grpcgrp 18492  inv_gcminusg 18493  -_gcsg 18494  1_rcur 19652  Ringcrg 19698  inv_rcinvr 19828  DivRingcdr 19906  LModclmod 20038  LSubSpclss 20108  LSpanclspn 20148  LVecclvec 20279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-cntz 18838  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-drng 19908  df-lmod 20040  df-lss 20109  df-lsp 20149  df-lvec 20280

This theorem is referenced by:  lspexchn1  20307  lspindp1  20310  mapdh8ab  39718  mapdh8ad  39720  mapdh8b  39721  mapdh8c  39722  mapdh8e  39725