MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lspexch Structured version   Visualization version   GIF version
Theorem lspexch 20742
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 20743 versus lspexchn2 20744); look for lspexch 20742 and prcom 4737 in same proof. TODO: would a hypothesis of Β¬ 𝑋 ∈ (π‘β€˜{𝑍}) instead of (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}) be better overall? This would be shorter and also satisfy the 𝑋 β‰  0 condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the β‰  pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)
Hypotheses
Ref Expression
lspexch.v 𝑉 = (Baseβ€˜π‘Š)
lspexch.o 0 = (0gβ€˜π‘Š)
lspexch.n 𝑁 = (LSpanβ€˜π‘Š)
lspexch.w (πœ‘ β†’ π‘Š ∈ LVec)
lspexch.x (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
lspexch.y (πœ‘ β†’ π‘Œ ∈ 𝑉)
lspexch.z (πœ‘ β†’ 𝑍 ∈ 𝑉)
lspexch.q (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
lspexch.e (πœ‘ β†’ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
Assertion
Ref Expression
lspexch (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍}))
Proof of Theorem lspexch
Dummy variables 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lspexch.e . . 3 (πœ‘ β†’ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
2 lspexch.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
3 eqid 2733 . . . 4 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
4 eqid 2733 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2733 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
6 eqid 2733 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
7 lspexch.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
8 lspexch.w . . . . 5 (πœ‘ β†’ π‘Š ∈ LVec)
9 lveclmod 20717 . . . . 5 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
108, 9syl 17 . . . 4 (πœ‘ β†’ π‘Š ∈ LMod)
11 lspexch.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝑉)
12 lspexch.z . . . 4 (πœ‘ β†’ 𝑍 ∈ 𝑉)
132, 3, 4, 5, 6, 7, 10, 11, 12lspprel 20705 . . 3 (πœ‘ β†’ (𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}) ↔ βˆƒπ‘— ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆƒπ‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))))
141, 13mpbid 231 . 2 (πœ‘ β†’ βˆƒπ‘— ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆƒπ‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)))
15 eqid 2733 . . . . . . . 8 (-gβ€˜π‘Š) = (-gβ€˜π‘Š)
16 eqid 2733 . . . . . . . 8 (invgβ€˜(Scalarβ€˜π‘Š)) = (invgβ€˜(Scalarβ€˜π‘Š))
1783ad2ant1 1134 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ π‘Š ∈ LVec)
1817, 9syl 17 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ π‘Š ∈ LMod)
19 simp2r 1201 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
20 lspexch.x . . . . . . . . . 10 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
21203ad2ant1 1134 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
2221eldifad 3961 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ 𝑋 ∈ 𝑉)
23123ad2ant1 1134 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ 𝑍 ∈ 𝑉)
242, 3, 15, 6, 4, 5, 16, 18, 19, 22, 23lmodsubvs 20528 . . . . . . 7 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (𝑋(-gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)))
25 simp3 1139 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)))
2625eqcomd 2739 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = 𝑋)
27103ad2ant1 1134 . . . . . . . . . 10 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ π‘Š ∈ LMod)
28 lmodgrp 20478 . . . . . . . . . 10 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
2927, 28syl 17 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ π‘Š ∈ Grp)
302, 4, 6, 5lmodvscl 20489 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑍 ∈ 𝑉) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑍) ∈ 𝑉)
3118, 19, 23, 30syl3anc 1372 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑍) ∈ 𝑉)
32 simp2l 1200 . . . . . . . . . 10 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ 𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
33113ad2ant1 1134 . . . . . . . . . 10 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ π‘Œ ∈ 𝑉)
342, 4, 6, 5lmodvscl 20489 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘Œ ∈ 𝑉) β†’ (𝑗( ·𝑠 β€˜π‘Š)π‘Œ) ∈ 𝑉)
3518, 32, 33, 34syl3anc 1372 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (𝑗( ·𝑠 β€˜π‘Š)π‘Œ) ∈ 𝑉)
362, 3, 15grpsubadd 18911 . . . . . . . . 9 ((π‘Š ∈ Grp ∧ (𝑋 ∈ 𝑉 ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑍) ∈ 𝑉 ∧ (𝑗( ·𝑠 β€˜π‘Š)π‘Œ) ∈ 𝑉)) β†’ ((𝑋(-gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (𝑗( ·𝑠 β€˜π‘Š)π‘Œ) ↔ ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = 𝑋))
3729, 22, 31, 35, 36syl13anc 1373 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ((𝑋(-gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (𝑗( ·𝑠 β€˜π‘Š)π‘Œ) ↔ ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = 𝑋))
3826, 37mpbird 257 . . . . . . 7 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (𝑋(-gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (𝑗( ·𝑠 β€˜π‘Š)π‘Œ))
3924, 38eqtr3d 2775 . . . . . 6 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)) = (𝑗( ·𝑠 β€˜π‘Š)π‘Œ))
40 eqid 2733 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
41 eqid 2733 . . . . . . 7 (invrβ€˜(Scalarβ€˜π‘Š)) = (invrβ€˜(Scalarβ€˜π‘Š))
42 lspexch.q . . . . . . . . . 10 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
43423ad2ant1 1134 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
44 lspexch.o . . . . . . . . . . . 12 0 = (0gβ€˜π‘Š)
4517adantr 482 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ π‘Š ∈ LVec)
4623adantr 482 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ 𝑍 ∈ 𝑉)
4725adantr 482 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)))
48 oveq1 7416 . . . . . . . . . . . . . . . 16 (𝑗 = (0gβ€˜(Scalarβ€˜π‘Š)) β†’ (𝑗( ·𝑠 β€˜π‘Š)π‘Œ) = ((0gβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ))
4948oveq1d 7424 . . . . . . . . . . . . . . 15 (𝑗 = (0gβ€˜(Scalarβ€˜π‘Š)) β†’ ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (((0gβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)))
502, 4, 6, 40, 44lmod0vs 20505 . . . . . . . . . . . . . . . . . 18 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ) = 0 )
5118, 33, 50syl2anc 585 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ) = 0 )
5251oveq1d 7424 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (((0gβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = ( 0 (+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)))
532, 3, 44lmod0vlid 20502 . . . . . . . . . . . . . . . . 17 ((π‘Š ∈ LMod ∧ (π‘˜( ·𝑠 β€˜π‘Š)𝑍) ∈ 𝑉) β†’ ( 0 (+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (π‘˜( ·𝑠 β€˜π‘Š)𝑍))
5418, 31, 53syl2anc 585 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ( 0 (+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (π‘˜( ·𝑠 β€˜π‘Š)𝑍))
5552, 54eqtrd 2773 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (((0gβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (π‘˜( ·𝑠 β€˜π‘Š)𝑍))
5649, 55sylan9eqr 2795 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) = (π‘˜( ·𝑠 β€˜π‘Š)𝑍))
5747, 56eqtrd 2773 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ 𝑋 = (π‘˜( ·𝑠 β€˜π‘Š)𝑍))
582, 6, 4, 5, 7, 18, 19, 23lspsneli 20612 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑍) ∈ (π‘β€˜{𝑍}))
5958adantr 482 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑍) ∈ (π‘β€˜{𝑍}))
6057, 59eqeltrd 2834 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ 𝑋 ∈ (π‘β€˜{𝑍}))
61 eldifsni 4794 . . . . . . . . . . . . . 14 (𝑋 ∈ (𝑉 βˆ– { 0 }) β†’ 𝑋 β‰  0 )
6221, 61syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ 𝑋 β‰  0 )
6362adantr 482 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ 𝑋 β‰  0 )
642, 44, 7, 45, 46, 60, 63lspsneleq 20728 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) ∧ 𝑗 = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ (π‘β€˜{𝑋}) = (π‘β€˜{𝑍}))
6564ex 414 . . . . . . . . . 10 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (𝑗 = (0gβ€˜(Scalarβ€˜π‘Š)) β†’ (π‘β€˜{𝑋}) = (π‘β€˜{𝑍})))
6665necon3d 2962 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}) β†’ 𝑗 β‰  (0gβ€˜(Scalarβ€˜π‘Š))))
6743, 66mpd 15 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ 𝑗 β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
68 eldifsn 4791 . . . . . . . 8 (𝑗 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) ↔ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑗 β‰  (0gβ€˜(Scalarβ€˜π‘Š))))
6932, 67, 68sylanbrc 584 . . . . . . 7 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ 𝑗 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))
704lmodfgrp 20480 . . . . . . . . . . 11 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Grp)
7127, 70syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (Scalarβ€˜π‘Š) ∈ Grp)
725, 16grpinvcl 18872 . . . . . . . . . 10 (((Scalarβ€˜π‘Š) ∈ Grp ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
7371, 19, 72syl2anc 585 . . . . . . . . 9 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
742, 4, 6, 5lmodvscl 20489 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ ((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑍 ∈ 𝑉) β†’ (((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍) ∈ 𝑉)
7518, 73, 23, 74syl3anc 1372 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍) ∈ 𝑉)
762, 3lmodvacl 20486 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍) ∈ 𝑉) β†’ (𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)) ∈ 𝑉)
7718, 22, 75, 76syl3anc 1372 . . . . . . 7 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)) ∈ 𝑉)
782, 6, 4, 5, 40, 41, 17, 69, 77, 33lvecinv 20726 . . . . . 6 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ((𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)) = (𝑗( ·𝑠 β€˜π‘Š)π‘Œ) ↔ π‘Œ = (((invrβ€˜(Scalarβ€˜π‘Š))β€˜π‘—)( ·𝑠 β€˜π‘Š)(𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)))))
7939, 78mpbid 231 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ π‘Œ = (((invrβ€˜(Scalarβ€˜π‘Š))β€˜π‘—)( ·𝑠 β€˜π‘Š)(𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍))))
80 eqid 2733 . . . . . . 7 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
812, 80, 7, 18, 22, 23lspprcl 20589 . . . . . 6 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (π‘β€˜{𝑋, 𝑍}) ∈ (LSubSpβ€˜π‘Š))
824lvecdrng 20716 . . . . . . . 8 (π‘Š ∈ LVec β†’ (Scalarβ€˜π‘Š) ∈ DivRing)
8317, 82syl 17 . . . . . . 7 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (Scalarβ€˜π‘Š) ∈ DivRing)
845, 40, 41drnginvrcl 20379 . . . . . . 7 (((Scalarβ€˜π‘Š) ∈ DivRing ∧ 𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑗 β‰  (0gβ€˜(Scalarβ€˜π‘Š))) β†’ ((invrβ€˜(Scalarβ€˜π‘Š))β€˜π‘—) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
8583, 32, 67, 84syl3anc 1372 . . . . . 6 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ((invrβ€˜(Scalarβ€˜π‘Š))β€˜π‘—) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
86 eqid 2733 . . . . . . . . . 10 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
872, 4, 6, 86lmodvs1 20500 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋) = 𝑋)
8818, 22, 87syl2anc 585 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋) = 𝑋)
8988oveq1d 7424 . . . . . . 7 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋)(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)) = (𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)))
904lmodring 20479 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Ring)
915, 86ringidcl 20083 . . . . . . . . 9 ((Scalarβ€˜π‘Š) ∈ Ring β†’ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
9218, 90, 913syl 18 . . . . . . . 8 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
932, 3, 6, 4, 5, 7, 18, 92, 73, 22, 23lsppreli 20701 . . . . . . 7 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋)(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)) ∈ (π‘β€˜{𝑋, 𝑍}))
9489, 93eqeltrrd 2835 . . . . . 6 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)) ∈ (π‘β€˜{𝑋, 𝑍}))
954, 6, 5, 80lssvscl 20566 . . . . . 6 (((π‘Š ∈ LMod ∧ (π‘β€˜{𝑋, 𝑍}) ∈ (LSubSpβ€˜π‘Š)) ∧ (((invrβ€˜(Scalarβ€˜π‘Š))β€˜π‘—) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ (𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍)) ∈ (π‘β€˜{𝑋, 𝑍}))) β†’ (((invrβ€˜(Scalarβ€˜π‘Š))β€˜π‘—)( ·𝑠 β€˜π‘Š)(𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍))) ∈ (π‘β€˜{𝑋, 𝑍}))
9618, 81, 85, 94, 95syl22anc 838 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ (((invrβ€˜(Scalarβ€˜π‘Š))β€˜π‘—)( ·𝑠 β€˜π‘Š)(𝑋(+gβ€˜π‘Š)(((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜)( ·𝑠 β€˜π‘Š)𝑍))) ∈ (π‘β€˜{𝑋, 𝑍}))
9779, 96eqeltrd 2834 . . . 4 ((πœ‘ ∧ (𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍))) β†’ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍}))
98973exp 1120 . . 3 (πœ‘ β†’ ((𝑗 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) β†’ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍}))))
9998rexlimdvv 3211 . 2 (πœ‘ β†’ (βˆƒπ‘— ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆƒπ‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))𝑋 = ((𝑗( ·𝑠 β€˜π‘Š)π‘Œ)(+gβ€˜π‘Š)(π‘˜( ·𝑠 β€˜π‘Š)𝑍)) β†’ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍})))
10014, 99mpd 15 1 (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071   βˆ– cdif 3946  {csn 4629  {cpr 4631  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  Grpcgrp 18819  invgcminusg 18820  -gcsg 18821  1rcur 20004  Ringcrg 20056  invrcinvr 20201  DivRingcdr 20357  LModclmod 20471  LSubSpclss 20542  LSpanclspn 20582  LVecclvec 20713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-cntz 19181  df-lsm 19504  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-drng 20359  df-lmod 20473  df-lss 20543  df-lsp 20583  df-lvec 20714
This theorem is referenced by:  lspexchn1  20743  lspindp1  20746  mapdh8ab  40696  mapdh8ad  40698  mapdh8b  40699  mapdh8c  40700  mapdh8e  40703
  Copyright terms: Public domain W3C validator