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Theorem lhpset 34299
Description: The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
lhpset.b 𝐵 = (Base‘𝐾)
lhpset.u 1 = (1.‘𝐾)
lhpset.c 𝐶 = ( ⋖ ‘𝐾)
lhpset.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
lhpset (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
Distinct variable groups:   𝑤,𝐵   𝑤,𝐶   𝑤,𝐾   𝑤, 1
Allowed substitution hints:   𝐴(𝑤)   𝐻(𝑤)

Proof of Theorem lhpset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐾𝐴𝐾 ∈ V)
2 lhpset.h . . 3 𝐻 = (LHyp‘𝐾)
3 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lhpset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2662 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 eqidd 2611 . . . . . 6 (𝑘 = 𝐾𝑤 = 𝑤)
7 fveq2 6103 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
8 lhpset.c . . . . . . 7 𝐶 = ( ⋖ ‘𝐾)
97, 8syl6eqr 2662 . . . . . 6 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
10 fveq2 6103 . . . . . . 7 (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
11 lhpset.u . . . . . . 7 1 = (1.‘𝐾)
1210, 11syl6eqr 2662 . . . . . 6 (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
136, 9, 12breq123d 4597 . . . . 5 (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 ))
145, 13rabeqbidv 3168 . . . 4 (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤𝐵𝑤𝐶 1 })
15 df-lhyp 34292 . . . 4 LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)})
16 fvex 6113 . . . . . 6 (Base‘𝐾) ∈ V
174, 16eqeltri 2684 . . . . 5 𝐵 ∈ V
1817rabex 4740 . . . 4 {𝑤𝐵𝑤𝐶 1 } ∈ V
1914, 15, 18fvmpt 6191 . . 3 (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤𝐵𝑤𝐶 1 })
202, 19syl5eq 2656 . 2 (𝐾 ∈ V → 𝐻 = {𝑤𝐵𝑤𝐶 1 })
211, 20syl 17 1 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583  cfv 5804  Basecbs 15695  1.cp1 16861  ccvr 33567  LHypclh 34288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-lhyp 34292
This theorem is referenced by:  islhp  34300
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