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

Theorem lsatlspsn 37484
Description: The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
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)
lsatlspsn.w (πœ‘ β†’ π‘Š ∈ LMod)
lsatlspsn.x (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
Assertion
Ref Expression
lsatlspsn (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ 𝐴)

Proof of Theorem lsatlspsn
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 lsatlspsn.x . . 3 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
2 eqid 2737 . . 3 (π‘β€˜{𝑋}) = (π‘β€˜{𝑋})
3 sneq 4601 . . . . 5 (𝑣 = 𝑋 β†’ {𝑣} = {𝑋})
43fveq2d 6851 . . . 4 (𝑣 = 𝑋 β†’ (π‘β€˜{𝑣}) = (π‘β€˜{𝑋}))
54rspceeqv 3600 . . 3 ((𝑋 ∈ (𝑉 βˆ– { 0 }) ∧ (π‘β€˜{𝑋}) = (π‘β€˜{𝑋})) β†’ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣}))
61, 2, 5sylancl 587 . 2 (πœ‘ β†’ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣}))
7 lsatlspsn.w . . 3 (πœ‘ β†’ π‘Š ∈ LMod)
8 lsatset.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
9 lsatset.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
10 lsatset.z . . . 4 0 = (0gβ€˜π‘Š)
11 lsatset.a . . . 4 𝐴 = (LSAtomsβ€˜π‘Š)
128, 9, 10, 11islsat 37482 . . 3 (π‘Š ∈ LMod β†’ ((π‘β€˜{𝑋}) ∈ 𝐴 ↔ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣})))
137, 12syl 17 . 2 (πœ‘ β†’ ((π‘β€˜{𝑋}) ∈ 𝐴 ↔ βˆƒπ‘£ ∈ (𝑉 βˆ– { 0 })(π‘β€˜{𝑋}) = (π‘β€˜{𝑣})))
146, 13mpbird 257 1 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  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:  lsatspn0  37491  dvh4dimlem  39935  dochsnshp  39945  lclkrlem2a  39999  lclkrlem2c  40001  lclkrlem2e  40003  lcfrlem20  40054  mapdrvallem2  40137  mapdpglem20  40183  mapdpglem30a  40187  mapdpglem30b  40188  hdmaprnlem3eN  40350  hdmaprnlem16N  40354
  Copyright terms: Public domain W3C validator