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Theorem lspexch 19894
 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 19895 versus lspexchn2 19896); look for lspexch 19894 and prcom 4628 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 = (0g𝑊)
lspexch.n 𝑁 = (LSpan‘𝑊)
lspexch.w (𝜑𝑊 ∈ LVec)
lspexch.x (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
lspexch.y (𝜑𝑌𝑉)
lspexch.z (𝜑𝑍𝑉)
lspexch.q (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
lspexch.e (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
Assertion
Ref Expression
lspexch (𝜑𝑌 ∈ (𝑁‘{𝑋, 𝑍}))

Proof of Theorem lspexch
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lspexch.e . . 3 (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
2 lspexch.v . . . 4 𝑉 = (Base‘𝑊)
3 eqid 2798 . . . 4 (+g𝑊) = (+g𝑊)
4 eqid 2798 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2798 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2798 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 lspexch.n . . . 4 𝑁 = (LSpan‘𝑊)
8 lspexch.w . . . . 5 (𝜑𝑊 ∈ LVec)
9 lveclmod 19871 . . . . 5 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
108, 9syl 17 . . . 4 (𝜑𝑊 ∈ LMod)
11 lspexch.y . . . 4 (𝜑𝑌𝑉)
12 lspexch.z . . . 4 (𝜑𝑍𝑉)
132, 3, 4, 5, 6, 7, 10, 11, 12lspprel 19859 . . 3 (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))))
141, 13mpbid 235 . 2 (𝜑 → ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)))
15 eqid 2798 . . . . . . . 8 (-g𝑊) = (-g𝑊)
16 eqid 2798 . . . . . . . 8 (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊))
1783ad2ant1 1130 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LVec)
1817, 9syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
19 simp2r 1197 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
20 lspexch.x . . . . . . . . . 10 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
21203ad2ant1 1130 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
2221eldifad 3893 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋𝑉)
23123ad2ant1 1130 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑍𝑉)
242, 3, 15, 6, 4, 5, 16, 18, 19, 22, 23lmodsubvs 19683 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑋(-g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)))
25 simp3 1135 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)))
2625eqcomd 2804 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = 𝑋)
27103ad2ant1 1130 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
28 lmodgrp 19634 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2927, 28syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑊 ∈ Grp)
302, 4, 6, 5lmodvscl 19644 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉)
3118, 19, 23, 30syl3anc 1368 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉)
32 simp2l 1196 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑗 ∈ (Base‘(Scalar‘𝑊)))
33113ad2ant1 1130 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌𝑉)
342, 4, 6, 5lmodvscl 19644 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉) → (𝑗( ·𝑠𝑊)𝑌) ∈ 𝑉)
3518, 32, 33, 34syl3anc 1368 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑗( ·𝑠𝑊)𝑌) ∈ 𝑉)
362, 3, 15grpsubadd 18179 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ (𝑋𝑉 ∧ (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉 ∧ (𝑗( ·𝑠𝑊)𝑌) ∈ 𝑉)) → ((𝑋(-g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑗( ·𝑠𝑊)𝑌) ↔ ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = 𝑋))
3729, 22, 31, 35, 36syl13anc 1369 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ((𝑋(-g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑗( ·𝑠𝑊)𝑌) ↔ ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = 𝑋))
3826, 37mpbird 260 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑋(-g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑗( ·𝑠𝑊)𝑌))
3924, 38eqtr3d 2835 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) = (𝑗( ·𝑠𝑊)𝑌))
40 eqid 2798 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
41 eqid 2798 . . . . . . 7 (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
42 lspexch.q . . . . . . . . . 10 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43423ad2ant1 1130 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
44 lspexch.o . . . . . . . . . . . 12 0 = (0g𝑊)
4517adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑊 ∈ LVec)
4623adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑍𝑉)
4725adantr 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)))
48 oveq1 7142 . . . . . . . . . . . . . . . 16 (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌))
4948oveq1d 7150 . . . . . . . . . . . . . . 15 (𝑗 = (0g‘(Scalar‘𝑊)) → ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)))
502, 4, 6, 40, 44lmod0vs 19660 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
5118, 33, 50syl2anc 587 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
5251oveq1d 7150 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = ( 0 (+g𝑊)(𝑘( ·𝑠𝑊)𝑍)))
532, 3, 44lmod0vlid 19657 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉) → ( 0 (+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑘( ·𝑠𝑊)𝑍))
5418, 31, 53syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ( 0 (+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑘( ·𝑠𝑊)𝑍))
5552, 54eqtrd 2833 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑘( ·𝑠𝑊)𝑍))
5649, 55sylan9eqr 2855 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑘( ·𝑠𝑊)𝑍))
5747, 56eqtrd 2833 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑋 = (𝑘( ·𝑠𝑊)𝑍))
582, 6, 4, 5, 7, 18, 19, 23lspsneli 19766 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
5958adantr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → (𝑘( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
6057, 59eqeltrd 2890 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑋 ∈ (𝑁‘{𝑍}))
61 eldifsni 4683 . . . . . . . . . . . . . 14 (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋0 )
6221, 61syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋0 )
6362adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑋0 )
642, 44, 7, 45, 46, 60, 63lspsneleq 19880 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → (𝑁‘{𝑋}) = (𝑁‘{𝑍}))
6564ex 416 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑁‘{𝑋}) = (𝑁‘{𝑍})))
6665necon3d 3008 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) → 𝑗 ≠ (0g‘(Scalar‘𝑊))))
6743, 66mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑗 ≠ (0g‘(Scalar‘𝑊)))
68 eldifsn 4680 . . . . . . . 8 (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
6932, 67, 68sylanbrc 586 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
704lmodfgrp 19636 . . . . . . . . . . 11 (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Grp)
7127, 70syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (Scalar‘𝑊) ∈ Grp)
725, 16grpinvcl 18143 . . . . . . . . . 10 (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
7371, 19, 72syl2anc 587 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
742, 4, 6, 5lmodvscl 19644 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉)
7518, 73, 23, 74syl3anc 1368 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉)
762, 3lmodvacl 19641 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑋𝑉 ∧ (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉) → (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) ∈ 𝑉)
7718, 22, 75, 76syl3anc 1368 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) ∈ 𝑉)
782, 6, 4, 5, 40, 41, 17, 69, 77, 33lvecinv 19878 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ((𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) = (𝑗( ·𝑠𝑊)𝑌) ↔ 𝑌 = (((invr‘(Scalar‘𝑊))‘𝑗)( ·𝑠𝑊)(𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)))))
7939, 78mpbid 235 . . . . 5 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌 = (((invr‘(Scalar‘𝑊))‘𝑗)( ·𝑠𝑊)(𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍))))
80 eqid 2798 . . . . . . 7 (LSubSp‘𝑊) = (LSubSp‘𝑊)
812, 80, 7, 18, 22, 23lspprcl 19743 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑋, 𝑍}) ∈ (LSubSp‘𝑊))
824lvecdrng 19870 . . . . . . . 8 (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
8317, 82syl 17 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (Scalar‘𝑊) ∈ DivRing)
845, 40, 41drnginvrcl 19512 . . . . . . 7 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)))
8583, 32, 67, 84syl3anc 1368 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)))
86 eqid 2798 . . . . . . . . . 10 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
872, 4, 6, 86lmodvs1 19655 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑋) = 𝑋)
8818, 22, 87syl2anc 587 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑋) = 𝑋)
8988oveq1d 7150 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑋)(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) = (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)))
904lmodring 19635 . . . . . . . . 9 (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
915, 86ringidcl 19314 . . . . . . . . 9 ((Scalar‘𝑊) ∈ Ring → (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
9218, 90, 913syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
932, 3, 6, 4, 5, 7, 18, 92, 73, 22, 23lsppreli 19855 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑋)(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) ∈ (𝑁‘{𝑋, 𝑍}))
9489, 93eqeltrrd 2891 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) ∈ (𝑁‘{𝑋, 𝑍}))
954, 6, 5, 80lssvscl 19720 . . . . . 6 (((𝑊 ∈ LMod ∧ (𝑁‘{𝑋, 𝑍}) ∈ (LSubSp‘𝑊)) ∧ (((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) ∈ (𝑁‘{𝑋, 𝑍}))) → (((invr‘(Scalar‘𝑊))‘𝑗)( ·𝑠𝑊)(𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍))) ∈ (𝑁‘{𝑋, 𝑍}))
9618, 81, 85, 94, 95syl22anc 837 . . . . 5 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑗)( ·𝑠𝑊)(𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍))) ∈ (𝑁‘{𝑋, 𝑍}))
9779, 96eqeltrd 2890 . . . 4 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))
98973exp 1116 . . 3 (𝜑 → ((𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))))
9998rexlimdvv 3252 . 2 (𝜑 → (∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
10014, 99mpd 15 1 (𝜑𝑌 ∈ (𝑁‘{𝑋, 𝑍}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107   ∖ cdif 3878  {csn 4525  {cpr 4527  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  Scalarcsca 16560   ·𝑠 cvsca 16561  0gc0g 16705  Grpcgrp 18095  invgcminusg 18096  -gcsg 18097  1rcur 19244  Ringcrg 19290  invrcinvr 19417  DivRingcdr 19495  LModclmod 19627  LSubSpclss 19696  LSpanclspn 19736  LVecclvec 19867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-tpos 7875  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-cntz 18439  df-lsm 18753  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-oppr 19369  df-dvdsr 19387  df-unit 19388  df-invr 19418  df-drng 19497  df-lmod 19629  df-lss 19697  df-lsp 19737  df-lvec 19868 This theorem is referenced by:  lspexchn1  19895  lspindp1  19898  mapdh8ab  39070  mapdh8ad  39072  mapdh8b  39073  mapdh8c  39074  mapdh8e  39077
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