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Theorem lhpset 38861
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 3492 . 2 (𝐾𝐴𝐾 ∈ V)
2 lhpset.h . . 3 𝐻 = (LHyp‘𝐾)
3 fveq2 6891 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lhpset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2790 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 eqidd 2733 . . . . . 6 (𝑘 = 𝐾𝑤 = 𝑤)
7 fveq2 6891 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
8 lhpset.c . . . . . . 7 𝐶 = ( ⋖ ‘𝐾)
97, 8eqtr4di 2790 . . . . . 6 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
10 fveq2 6891 . . . . . . 7 (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
11 lhpset.u . . . . . . 7 1 = (1.‘𝐾)
1210, 11eqtr4di 2790 . . . . . 6 (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
136, 9, 12breq123d 5162 . . . . 5 (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 ))
145, 13rabeqbidv 3449 . . . 4 (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤𝐵𝑤𝐶 1 })
15 df-lhyp 38854 . . . 4 LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)})
164fvexi 6905 . . . . 5 𝐵 ∈ V
1716rabex 5332 . . . 4 {𝑤𝐵𝑤𝐶 1 } ∈ V
1814, 15, 17fvmpt 6998 . . 3 (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤𝐵𝑤𝐶 1 })
192, 18eqtrid 2784 . 2 (𝐾 ∈ V → 𝐻 = {𝑤𝐵𝑤𝐶 1 })
201, 19syl 17 1 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {crab 3432  Vcvv 3474   class class class wbr 5148  cfv 6543  Basecbs 17143  1.cp1 18376  ccvr 38127  LHypclh 38850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-lhyp 38854
This theorem is referenced by:  islhp  38862
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