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Theorem lhpset 38504
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 3462 . 2 (𝐾𝐴𝐾 ∈ V)
2 lhpset.h . . 3 𝐻 = (LHyp‘𝐾)
3 fveq2 6843 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lhpset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2791 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 eqidd 2734 . . . . . 6 (𝑘 = 𝐾𝑤 = 𝑤)
7 fveq2 6843 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
8 lhpset.c . . . . . . 7 𝐶 = ( ⋖ ‘𝐾)
97, 8eqtr4di 2791 . . . . . 6 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
10 fveq2 6843 . . . . . . 7 (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
11 lhpset.u . . . . . . 7 1 = (1.‘𝐾)
1210, 11eqtr4di 2791 . . . . . 6 (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
136, 9, 12breq123d 5120 . . . . 5 (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 ))
145, 13rabeqbidv 3423 . . . 4 (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤𝐵𝑤𝐶 1 })
15 df-lhyp 38497 . . . 4 LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)})
164fvexi 6857 . . . . 5 𝐵 ∈ V
1716rabex 5290 . . . 4 {𝑤𝐵𝑤𝐶 1 } ∈ V
1814, 15, 17fvmpt 6949 . . 3 (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤𝐵𝑤𝐶 1 })
192, 18eqtrid 2785 . 2 (𝐾 ∈ V → 𝐻 = {𝑤𝐵𝑤𝐶 1 })
201, 19syl 17 1 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3444   class class class wbr 5106  cfv 6497  Basecbs 17088  1.cp1 18318  ccvr 37770  LHypclh 38493
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 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-lhyp 38497
This theorem is referenced by:  islhp  38505
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