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Theorem lsatlspsn2 37483
Description: The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 37484 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 1151 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ) β†’ (𝑋 ∈ 𝑉 ∧ 𝑋 β‰  0 ))
2 eldifsn 4752 . . . 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 2737 . . 3 (π‘β€˜{𝑋}) = (π‘β€˜{𝑋})
5 sneq 4601 . . . . 5 (𝑣 = 𝑋 β†’ {𝑣} = {𝑋})
65fveq2d 6851 . . . 4 (𝑣 = 𝑋 β†’ (π‘β€˜{𝑣}) = (π‘β€˜{𝑋}))
76rspceeqv 3600 . . 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 587 . 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 37482 . . 3 (π‘Š ∈ LMod β†’ ((π‘β€˜{𝑋}) ∈ 𝐴 ↔ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣})))
14133ad2ant1 1134 . 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 257 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   βˆ– cdif 3912  {csn 4591  β€˜cfv 6501  Basecbs 17090  0gc0g 17328  LModclmod 20338  LSpanclspn 20448  LSAtomsclsa 37465
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-lsatoms 37467
This theorem is referenced by:  lsatel  37496  lsmsat  37499  lssatomic  37502  lssats  37503  dihlsprn  39823  dihatlat  39826  dihatexv  39830  dochsatshpb  39944
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