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Mirrors  >  Home  >  MPE Home  >  Th. List  >  lspexch Structured version   Visualization version   GIF version
Theorem lspexch 20734
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 20735 versus lspexchn2 20736); look for lspexch 20734 and prcom 4735 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 2732 . . . 4 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
4 eqid 2732 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2732 . . . 4 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
6 eqid 2732 . . . 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 20709 . . . . 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 20697 . . 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 2732 . . . . . . . 8 (-gβ€˜π‘Š) = (-gβ€˜π‘Š)
16 eqid 2732 . . . . . . . 8 (invgβ€˜(Scalarβ€˜π‘Š)) = (invgβ€˜(Scalarβ€˜π‘Š))
1783ad2ant1 1133 . . . . . . . . 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 1200 . . . . . . . 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 1133 . . . . . . . . 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 3959 . . . . . . . 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 1133 . . . . . . . 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 20520 . . . . . . 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 1138 . . . . . . . . 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 2738 . . . . . . . 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 1133 . . . . . . . . . 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 20470 . . . . . . . . . 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 20481 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑍 ∈ 𝑉) β†’ (π‘˜( ·𝑠 β€˜π‘Š)𝑍) ∈ 𝑉)
3118, 19, 23, 30syl3anc 1371 . . . . . . . . 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 1199 . . . . . . . . . 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 1133 . . . . . . . . . 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 20481 . . . . . . . . . 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 1371 . . . . . . . . 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 18907 . . . . . . . . 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 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†’ ((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 256 . . . . . . 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 2774 . . . . . 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 2732 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
41 eqid 2732 . . . . . . 7 (invrβ€˜(Scalarβ€˜π‘Š)) = (invrβ€˜(Scalarβ€˜π‘Š))
42 lspexch.q . . . . . . . . . 10 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
43423ad2ant1 1133 . . . . . . . . 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 481 . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . . 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 20497 . . . . . . . . . . . . . . . . . 18 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ) = 0 )
5118, 33, 50syl2anc 584 . . . . . . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . . . 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 20494 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . 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 20604 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 2833 . . . . . . . . . . . 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 4792 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 20720 . . . . . . . . . . 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 413 . . . . . . . . . 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 2961 . . . . . . . . 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 4789 . . . . . . . 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 583 . . . . . . 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 20472 . . . . . . . . . . 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 18868 . . . . . . . . . 10 (((Scalarβ€˜π‘Š) ∈ Grp ∧ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ((invgβ€˜(Scalarβ€˜π‘Š))β€˜π‘˜) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
7371, 19, 72syl2anc 584 . . . . . . . . 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 20481 . . . . . . . . 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 1371 . . . . . . . 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 20478 . . . . . . . 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 1371 . . . . . . 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 20718 . . . . . 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 2732 . . . . . . 7 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
812, 80, 7, 18, 22, 23lspprcl 20581 . . . . . 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 20708 . . . . . . . 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 20329 . . . . . . 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 1371 . . . . . 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 2732 . . . . . . . . . 10 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
872, 4, 6, 86lmodvs1 20492 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋) = 𝑋)
8818, 22, 87syl2anc 584 . . . . . . . 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 7420 . . . . . . 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 20471 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Ring)
915, 86ringidcl 20076 . . . . . . . . 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 20693 . . . . . . 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 2834 . . . . . 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 20558 . . . . . 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 837 . . . . 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 2833 . . . 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 1119 . . 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 3210 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   βˆ– cdif 3944  {csn 4627  {cpr 4629  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  Grpcgrp 18815  invgcminusg 18816  -gcsg 18817  1rcur 19998  Ringcrg 20049  invrcinvr 20193  DivRingcdr 20307  LModclmod 20463  LSubSpclss 20534  LSpanclspn 20574  LVecclvec 20705
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-subg 18997  df-cntz 19175  df-lsm 19498  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-oppr 20142  df-dvdsr 20163  df-unit 20164  df-invr 20194  df-drng 20309  df-lmod 20465  df-lss 20535  df-lsp 20575  df-lvec 20706
This theorem is referenced by:  lspexchn1  20735  lspindp1  20738  mapdh8ab  40636  mapdh8ad  40638  mapdh8b  40639  mapdh8c  40640  mapdh8e  40643
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