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Theorem lsatlspsn 38376
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 2726 . . 3 (π‘β€˜{𝑋}) = (π‘β€˜{𝑋})
3 sneq 4633 . . . . 5 (𝑣 = 𝑋 β†’ {𝑣} = {𝑋})
43fveq2d 6889 . . . 4 (𝑣 = 𝑋 β†’ (π‘β€˜{𝑣}) = (π‘β€˜{𝑋}))
54rspceeqv 3628 . . 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 585 . 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 38374 . . 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 1533   ∈ wcel 2098  βˆƒwrex 3064   βˆ– cdif 3940  {csn 4623  β€˜cfv 6537  Basecbs 17153  0gc0g 17394  LModclmod 20706  LSpanclspn 20818  LSAtomsclsa 38357
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-lsatoms 38359
This theorem is referenced by:  lsatspn0  38383  dvh4dimlem  40827  dochsnshp  40837  lclkrlem2a  40891  lclkrlem2c  40893  lclkrlem2e  40895  lcfrlem20  40946  mapdrvallem2  41029  mapdpglem20  41075  mapdpglem30a  41079  mapdpglem30b  41080  hdmaprnlem3eN  41242  hdmaprnlem16N  41246
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