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Theorem lsatlspsn2 34597
Description: The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 34598 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lsatset.v 𝑉 = (Base‘𝑊)
lsatset.n 𝑁 = (LSpan‘𝑊)
lsatset.z 0 = (0g𝑊)
lsatset.a 𝐴 = (LSAtoms‘𝑊)
Assertion
Ref Expression
lsatlspsn2 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → (𝑁‘{𝑋}) ∈ 𝐴)

Proof of Theorem lsatlspsn2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 3simpc 1080 . . . 4 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → (𝑋𝑉𝑋0 ))
2 eldifsn 4350 . . . 4 (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋𝑉𝑋0 ))
31, 2sylibr 224 . . 3 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
4 eqid 2651 . . 3 (𝑁‘{𝑋}) = (𝑁‘{𝑋})
5 sneq 4220 . . . . . 6 (𝑣 = 𝑋 → {𝑣} = {𝑋})
65fveq2d 6233 . . . . 5 (𝑣 = 𝑋 → (𝑁‘{𝑣}) = (𝑁‘{𝑋}))
76eqeq2d 2661 . . . 4 (𝑣 = 𝑋 → ((𝑁‘{𝑋}) = (𝑁‘{𝑣}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑋})))
87rspcev 3340 . . 3 ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑁‘{𝑋}) = (𝑁‘{𝑋})) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))
93, 4, 8sylancl 695 . 2 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))
10 lsatset.v . . . 4 𝑉 = (Base‘𝑊)
11 lsatset.n . . . 4 𝑁 = (LSpan‘𝑊)
12 lsatset.z . . . 4 0 = (0g𝑊)
13 lsatset.a . . . 4 𝐴 = (LSAtoms‘𝑊)
1410, 11, 12, 13islsat 34596 . . 3 (𝑊 ∈ LMod → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})))
15143ad2ant1 1102 . 2 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})))
169, 15mpbird 247 1 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → (𝑁‘{𝑋}) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  cdif 3604  {csn 4210  cfv 5926  Basecbs 15904  0gc0g 16147  LModclmod 18911  LSpanclspn 19019  LSAtomsclsa 34579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-lsatoms 34581
This theorem is referenced by:  lsatel  34610  lsmsat  34613  lssatomic  34616  lssats  34617  dihlsprn  36937  dihatlat  36940  dihatexv  36944  dochsatshpb  37058
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