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Theorem lsatlspsn2 38166
Description: The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 38167 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lsatset.v 𝑉 = (Baseβ€˜π‘Š)
lsatset.n 𝑁 = (LSpanβ€˜π‘Š)
lsatset.z 0 = (0gβ€˜π‘Š)
lsatset.a 𝐴 = (LSAtomsβ€˜π‘Š)
Assertion
Ref Expression
lsatlspsn2 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ) β†’ (π‘β€˜{𝑋}) ∈ 𝐴)

Proof of Theorem lsatlspsn2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 3simpc 1149 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ) β†’ (𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ))
2 eldifsn 4791 . . . 4 (𝑋 ∈ (𝑉 βˆ– { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ))
31, 2sylibr 233 . . 3 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ) β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
4 eqid 2731 . . 3 (π‘β€˜{𝑋}) = (π‘β€˜{𝑋})
5 sneq 4639 . . . . 5 (𝑣 = 𝑋 β†’ {𝑣} = {𝑋})
65fveq2d 6896 . . . 4 (𝑣 = 𝑋 β†’ (π‘β€˜{𝑣}) = (π‘β€˜{𝑋}))
76rspceeqv 3634 . . 3 ((𝑋 ∈ (𝑉 βˆ– { 0 }) ∧ (π‘β€˜{𝑋}) = (π‘β€˜{𝑋})) β†’ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣}))
83, 4, 7sylancl 585 . 2 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ) β†’ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣}))
9 lsatset.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
10 lsatset.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
11 lsatset.z . . . 4 0 = (0gβ€˜π‘Š)
12 lsatset.a . . . 4 𝐴 = (LSAtomsβ€˜π‘Š)
139, 10, 11, 12islsat 38165 . . 3 (π‘Š ∈ LMod β†’ ((π‘β€˜{𝑋}) ∈ 𝐴 ↔ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣})))
14133ad2ant1 1132 . 2 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ) β†’ ((π‘β€˜{𝑋}) ∈ 𝐴 ↔ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣})))
158, 14mpbird 256 1 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ) β†’ (π‘β€˜{𝑋}) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069   βˆ– cdif 3946  {csn 4629  β€˜cfv 6544  Basecbs 17149  0gc0g 17390  LModclmod 20615  LSpanclspn 20727  LSAtomsclsa 38148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  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-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-lsatoms 38150
This theorem is referenced by:  lsatel  38179  lsmsat  38182  lssatomic  38185  lssats  38186  dihlsprn  40506  dihatlat  40509  dihatexv  40513  dochsatshpb  40627
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