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Theorem lspexch 19342
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 19343 vs. lspexchn2 19344); look for lspexch 19342 and prcom 4401 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 2770 . . . 4 (+g𝑊) = (+g𝑊)
4 eqid 2770 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2770 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2770 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 lspexch.n . . . 4 𝑁 = (LSpan‘𝑊)
8 lspexch.w . . . . 5 (𝜑𝑊 ∈ LVec)
9 lveclmod 19318 . . . . 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 19306 . . 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 2770 . . . . . . . 8 (-g𝑊) = (-g𝑊)
16 eqid 2770 . . . . . . . 8 (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊))
1783ad2ant1 1126 . . . . . . . . 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 1241 . . . . . . . 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 1126 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
2221eldifad 3733 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑋𝑉)
23123ad2ant1 1126 . . . . . . . 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 19128 . . . . . . 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 1131 . . . . . . . . 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 2776 . . . . . . . 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 1126 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
28 lmodgrp 19079 . . . . . . . . . 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 19089 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉)
3118, 19, 23, 30syl3anc 1475 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉)
32 simp2l 1240 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑗 ∈ (Base‘(Scalar‘𝑊)))
33113ad2ant1 1126 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌𝑉)
342, 4, 6, 5lmodvscl 19089 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉) → (𝑗( ·𝑠𝑊)𝑌) ∈ 𝑉)
3518, 32, 33, 34syl3anc 1475 . . . . . . . . 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 17710 . . . . . . . . 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 1477 . . . . . . . 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 2806 . . . . . 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 2770 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
41 eqid 2770 . . . . . . 7 (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
42 lspexch.q . . . . . . . . . 10 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
43423ad2ant1 1126 . . . . . . . . 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 466 . . . . . . . . . . . 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 466 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) ∧ 𝑗 = (0g‘(Scalar‘𝑊))) → 𝑍𝑉)
4725adantr 466 . . . . . . . . . . . . . 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 6799 . . . . . . . . . . . . . . . 16 (𝑗 = (0g‘(Scalar‘𝑊)) → (𝑗( ·𝑠𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌))
4948oveq1d 6807 . . . . . . . . . . . . . . 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 19105 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
5118, 33, 50syl2anc 565 . . . . . . . . . . . . . . . . 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 6807 . . . . . . . . . . . . . . . 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 19102 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑍) ∈ 𝑉) → ( 0 (+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) = (𝑘( ·𝑠𝑊)𝑍))
5418, 31, 53syl2anc 565 . . . . . . . . . . . . . . . 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 2804 . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . 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 2804 . . . . . . . . . . . . 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 19213 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
5958adantr 466 . . . . . . . . . . . . 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 2849 . . . . . . . . . . . 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 4455 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 19327 . . . . . . . . . . 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 397 . . . . . . . . . 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 2963 . . . . . . . . 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 4451 . . . . . . . 8 (𝑗 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ↔ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))))
6932, 67, 68sylanbrc 564 . . . . . . 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 19081 . . . . . . . . . . 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 17674 . . . . . . . . . 10 (((Scalar‘𝑊) ∈ Grp ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
7371, 19, 72syl2anc 565 . . . . . . . . 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 19089 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ ((invg‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉)
7518, 73, 23, 74syl3anc 1475 . . . . . . . 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 19086 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑋𝑉 ∧ (((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍) ∈ 𝑉) → (𝑋(+g𝑊)(((invg‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)𝑍)) ∈ 𝑉)
7718, 22, 75, 76syl3anc 1475 . . . . . . 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 19325 . . . . . 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 2770 . . . . . . 7 (LSubSp‘𝑊) = (LSubSp‘𝑊)
812, 80, 7, 18, 22, 23lspprcl 19190 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑋, 𝑍}) ∈ (LSubSp‘𝑊))
824lvecdrng 19317 . . . . . . . 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 18973 . . . . . . 7 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑗 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑗) ∈ (Base‘(Scalar‘𝑊)))
8583, 32, 67, 84syl3anc 1475 . . . . . 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 2770 . . . . . . . . . 10 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
872, 4, 6, 86lmodvs1 19100 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑋) = 𝑋)
8818, 22, 87syl2anc 565 . . . . . . . 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 6807 . . . . . . 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 19080 . . . . . . . . 9 (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
915, 86ringidcl 18775 . . . . . . . . 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 19302 . . . . . . 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 2850 . . . . . 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 19167 . . . . . 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 1476 . . . . 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 2849 . . . 4 ((𝜑 ∧ (𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))
98973exp 1111 . . 3 (𝜑 → ((𝑗 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑋 = ((𝑗( ·𝑠𝑊)𝑌)(+g𝑊)(𝑘( ·𝑠𝑊)𝑍)) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))))
9998rexlimdvv 3184 . 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 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wrex 3061  cdif 3718  {csn 4314  {cpr 4316  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  Scalarcsca 16151   ·𝑠 cvsca 16152  0gc0g 16307  Grpcgrp 17629  invgcminusg 17630  -gcsg 17631  1rcur 18708  Ringcrg 18754  invrcinvr 18878  DivRingcdr 18956  LModclmod 19072  LSubSpclss 19141  LSpanclspn 19183  LVecclvec 19314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-tpos 7503  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-submnd 17543  df-grp 17632  df-minusg 17633  df-sbg 17634  df-subg 17798  df-cntz 17956  df-lsm 18257  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-oppr 18830  df-dvdsr 18848  df-unit 18849  df-invr 18879  df-drng 18958  df-lmod 19074  df-lss 19142  df-lsp 19184  df-lvec 19315
This theorem is referenced by:  lspexchn1  19343  lspindp1  19346  mapdh8ab  37580  mapdh8ad  37582  mapdh8b  37583  mapdh8c  37584  mapdh8e  37587
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