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Theorem lspexch 19177
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 19178 vs. lspexchn2 19179); look for lspexch 19177 and prcom 4299 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 𝑋0 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 2651 . . . 4 (+g𝑊) = (+g𝑊)
4 eqid 2651 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2651 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2651 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 lspexch.n . . . 4 𝑁 = (LSpan‘𝑊)
8 lspexch.w . . . . 5 (𝜑𝑊 ∈ LVec)
9 lveclmod 19154 . . . . 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 19142 . . 3 (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))))
141, 13mpbid 222 . 2 (𝜑 → ∃𝑗 ∈ (Base‘(Scalar‘𝑊))∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)))
15 eqid 2651 . . . . . . . 8 (-g𝑊) = (-g𝑊)
16 eqid 2651 . . . . . . . 8 (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊))
1783ad2ant1 1102 . . . . . . . . 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 1108 . . . . . . . 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 1102 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
2221eldifad 3619 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋𝑉)
23123ad2ant1 1102 . . . . . . . 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 18967 . . . . . . 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 1083 . . . . . . . . 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 2657 . . . . . . . 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 1102 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
28 lmodgrp 18918 . . . . . . . . . 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 18928 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉)
3118, 19, 23, 30syl3anc 1366 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉)
32 simp2l 1107 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑗 ∈ (Base‘(Scalar‘𝑊)))
33113ad2ant1 1102 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌𝑉)
342, 4, 6, 5lmodvscl 18928 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉) → (𝑗( ·𝑠𝑊)𝑌) ∈ 𝑉)
3518, 32, 33, 34syl3anc 1366 . . . . . . . . 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 17550 . . . . . . . . 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 1368 . . . . . . . 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 247 . . . . . . 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 2687 . . . . . 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 2651 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
41 eqid 2651 . . . . . . 7 (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
42 lspexch.q . . . . . . . . . 10 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43423ad2ant1 1102 . . . . . . . . 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 480 . . . . . . . . . . . 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 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑍𝑉)
4725adantr 480 . . . . . . . . . . . . . 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 6697 . . . . . . . . . . . . . . . 16 (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌))
4948oveq1d 6705 . . . . . . . . . . . . . . 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 18944 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
5118, 33, 50syl2anc 694 . . . . . . . . . . . . . . . . 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 6705 . . . . . . . . . . . . . . . 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 18941 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉) → ( 0 (+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑘( ·𝑠𝑊)𝑍))
5418, 31, 53syl2anc 694 . . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . . 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 2707 . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . 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 19049 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
5958adantr 480 . . . . . . . . . . . . 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 2730 . . . . . . . . . . . 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 4353 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 19163 . . . . . . . . . . 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 449 . . . . . . . . . 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 2844 . . . . . . . . 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 4350 . . . . . . . 8 (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
6932, 67, 68sylanbrc 699 . . . . . . 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 18920 . . . . . . . . . . 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 17514 . . . . . . . . . 10 (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
7371, 19, 72syl2anc 694 . . . . . . . . 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 18928 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉)
7518, 73, 23, 74syl3anc 1366 . . . . . . . 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 18925 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑋𝑉 ∧ (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉) → (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) ∈ 𝑉)
7718, 22, 75, 76syl3anc 1366 . . . . . . 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 19161 . . . . . 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 222 . . . . 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 2651 . . . . . . 7 (LSubSp‘𝑊) = (LSubSp‘𝑊)
812, 80, 7, 18, 22, 23lspprcl 19026 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑋, 𝑍}) ∈ (LSubSp‘𝑊))
824lvecdrng 19153 . . . . . . . 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 18812 . . . . . . 7 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)))
8583, 32, 67, 84syl3anc 1366 . . . . . 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 2651 . . . . . . . . . 10 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
872, 4, 6, 86lmodvs1 18939 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑋) = 𝑋)
8818, 22, 87syl2anc 694 . . . . . . . 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 6705 . . . . . . 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 18919 . . . . . . . . 9 (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
915, 86ringidcl 18614 . . . . . . . . 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 19138 . . . . . . 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 2731 . . . . . 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 19003 . . . . . 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 1367 . . . . 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 2730 . . . 4 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))
98973exp 1283 . . 3 (𝜑 → ((𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))))
9998rexlimdvv 3066 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  cdif 3604  {csn 4210  {cpr 4212  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  Grpcgrp 17469  invgcminusg 17470  -gcsg 17471  1rcur 18547  Ringcrg 18593  invrcinvr 18717  DivRingcdr 18795  LModclmod 18911  LSubSpclss 18980  LSpanclspn 19019  LVecclvec 19150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-cntz 17796  df-lsm 18097  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-drng 18797  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lvec 19151
This theorem is referenced by:  lspexchn1  19178  lspindp1  19181  mapdh8ab  37383  mapdh8ad  37385  mapdh8b  37386  mapdh8c  37387  mapdh8e  37390
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