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Theorem lsatlspsn 38521
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 2725 . . 3 (π‘β€˜{𝑋}) = (π‘β€˜{𝑋})
3 sneq 4634 . . . . 5 (𝑣 = 𝑋 β†’ {𝑣} = {𝑋})
43fveq2d 6896 . . . 4 (𝑣 = 𝑋 β†’ (π‘β€˜{𝑣}) = (π‘β€˜{𝑋}))
54rspceeqv 3623 . . 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 584 . 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 38519 . . 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 256 1 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   βˆ– cdif 3936  {csn 4624  β€˜cfv 6543  Basecbs 17179  0gc0g 17420  LModclmod 20747  LSpanclspn 20859  LSAtomsclsa 38502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-lsatoms 38504
This theorem is referenced by:  lsatspn0  38528  dvh4dimlem  40972  dochsnshp  40982  lclkrlem2a  41036  lclkrlem2c  41038  lclkrlem2e  41040  lcfrlem20  41091  mapdrvallem2  41174  mapdpglem20  41220  mapdpglem30a  41224  mapdpglem30b  41225  hdmaprnlem3eN  41387  hdmaprnlem16N  41391
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