Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lsatlspsn2 Structured version   Visualization version   GIF version

Theorem lsatlspsn2 35060
Description: The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 35061 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 1186 . . . 4 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → (𝑋𝑉𝑋0 ))
2 eldifsn 4536 . . . 4 (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋𝑉𝑋0 ))
31, 2sylibr 226 . . 3 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
4 eqid 2825 . . 3 (𝑁‘{𝑋}) = (𝑁‘{𝑋})
5 sneq 4407 . . . . 5 (𝑣 = 𝑋 → {𝑣} = {𝑋})
65fveq2d 6437 . . . 4 (𝑣 = 𝑋 → (𝑁‘{𝑣}) = (𝑁‘{𝑋}))
76rspceeqv 3544 . . 3 ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑁‘{𝑋}) = (𝑁‘{𝑋})) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))
83, 4, 7sylancl 580 . 2 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))
9 lsatset.v . . . 4 𝑉 = (Base‘𝑊)
10 lsatset.n . . . 4 𝑁 = (LSpan‘𝑊)
11 lsatset.z . . . 4 0 = (0g𝑊)
12 lsatset.a . . . 4 𝐴 = (LSAtoms‘𝑊)
139, 10, 11, 12islsat 35059 . . 3 (𝑊 ∈ LMod → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})))
14133ad2ant1 1167 . 2 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})))
158, 14mpbird 249 1 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → (𝑁‘{𝑋}) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wrex 3118  cdif 3795  {csn 4397  cfv 6123  Basecbs 16222  0gc0g 16453  LModclmod 19219  LSpanclspn 19330  LSAtomsclsa 35042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-lsatoms 35044
This theorem is referenced by:  lsatel  35073  lsmsat  35076  lssatomic  35079  lssats  35080  dihlsprn  37399  dihatlat  37402  dihatexv  37406  dochsatshpb  37520
  Copyright terms: Public domain W3C validator