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Theorem lspexch 19043
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 19044 vs. lspexchn2 19045); look for lspexch 19043 and prcom 4242 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 2626 . . . 4 (+g𝑊) = (+g𝑊)
4 eqid 2626 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2626 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2626 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 lspexch.n . . . 4 𝑁 = (LSpan‘𝑊)
8 lspexch.w . . . . 5 (𝜑𝑊 ∈ LVec)
9 lveclmod 19020 . . . . 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 19008 . . 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 2626 . . . . . . . 8 (-g𝑊) = (-g𝑊)
16 eqid 2626 . . . . . . . 8 (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊))
1783ad2ant1 1080 . . . . . . . . 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 1086 . . . . . . . 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 1080 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
2221eldifad 3572 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋𝑉)
23123ad2ant1 1080 . . . . . . . 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 18835 . . . . . . 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 1061 . . . . . . . . 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 2632 . . . . . . . 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 1080 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
28 lmodgrp 18786 . . . . . . . . . 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 18796 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉)
3118, 19, 23, 30syl3anc 1323 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉)
32 simp2l 1085 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑗 ∈ (Base‘(Scalar‘𝑊)))
33113ad2ant1 1080 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌𝑉)
342, 4, 6, 5lmodvscl 18796 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉) → (𝑗( ·𝑠𝑊)𝑌) ∈ 𝑉)
3518, 32, 33, 34syl3anc 1323 . . . . . . . . 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 17419 . . . . . . . . 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 1325 . . . . . . . 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 2662 . . . . . 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 2626 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
41 eqid 2626 . . . . . . 7 (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
42 lspexch.q . . . . . . . . . 10 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43423ad2ant1 1080 . . . . . . . . 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 481 . . . . . . . . . . . 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 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑍𝑉)
4725adantr 481 . . . . . . . . . . . . . 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 6612 . . . . . . . . . . . . . . . 16 (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌))
4948oveq1d 6620 . . . . . . . . . . . . . . 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 18812 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
5118, 33, 50syl2anc 692 . . . . . . . . . . . . . . . . 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 6620 . . . . . . . . . . . . . . . 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 18809 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉) → ( 0 (+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑘( ·𝑠𝑊)𝑍))
5418, 31, 53syl2anc 692 . . . . . . . . . . . . . . . 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 2660 . . . . . . . . . . . . . . 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 2682 . . . . . . . . . . . . . 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 2660 . . . . . . . . . . . . 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 18915 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
5958adantr 481 . . . . . . . . . . . . 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 2704 . . . . . . . . . . . 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 4294 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 19029 . . . . . . . . . . 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 450 . . . . . . . . . 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 2817 . . . . . . . . 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 4292 . . . . . . . 8 (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
6932, 67, 68sylanbrc 697 . . . . . . 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 18788 . . . . . . . . . . 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 17383 . . . . . . . . . 10 (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
7371, 19, 72syl2anc 692 . . . . . . . . 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 18796 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉)
7518, 73, 23, 74syl3anc 1323 . . . . . . . 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 18793 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑋𝑉 ∧ (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉) → (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) ∈ 𝑉)
7718, 22, 75, 76syl3anc 1323 . . . . . . 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 19027 . . . . . 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 2626 . . . . . . 7 (LSubSp‘𝑊) = (LSubSp‘𝑊)
812, 80, 7, 18, 22, 23lspprcl 18892 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑋, 𝑍}) ∈ (LSubSp‘𝑊))
824lvecdrng 19019 . . . . . . . 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 18680 . . . . . . 7 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)))
8583, 32, 67, 84syl3anc 1323 . . . . . 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 2626 . . . . . . . . . 10 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
872, 4, 6, 86lmodvs1 18807 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑋) = 𝑋)
8818, 22, 87syl2anc 692 . . . . . . . 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 6620 . . . . . . 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 18787 . . . . . . . . 9 (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
915, 86ringidcl 18484 . . . . . . . . 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 19004 . . . . . . 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 2705 . . . . . 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 18869 . . . . . 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 1324 . . . . 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 2704 . . . 4 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))
98973exp 1261 . . 3 (𝜑 → ((𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))))
9998rexlimdvv 3035 . 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 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wrex 2913  cdif 3557  {csn 4153  {cpr 4155  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  Scalarcsca 15860   ·𝑠 cvsca 15861  0gc0g 16016  Grpcgrp 17338  invgcminusg 17339  -gcsg 17340  1rcur 18417  Ringcrg 18463  invrcinvr 18587  DivRingcdr 18663  LModclmod 18779  LSubSpclss 18846  LSpanclspn 18885  LVecclvec 19016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-subg 17507  df-cntz 17666  df-lsm 17967  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-oppr 18539  df-dvdsr 18557  df-unit 18558  df-invr 18588  df-drng 18665  df-lmod 18781  df-lss 18847  df-lsp 18886  df-lvec 19017
This theorem is referenced by:  lspexchn1  19044  lspindp1  19047  mapdh8ab  36532  mapdh8ad  36534  mapdh8b  36535  mapdh8c  36536  mapdh8e  36539
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