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Theorem lhpset 39472
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 3490 . 2 (𝐾𝐴𝐾 ∈ V)
2 lhpset.h . . 3 𝐻 = (LHyp‘𝐾)
3 fveq2 6900 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lhpset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2785 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 eqidd 2728 . . . . . 6 (𝑘 = 𝐾𝑤 = 𝑤)
7 fveq2 6900 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
8 lhpset.c . . . . . . 7 𝐶 = ( ⋖ ‘𝐾)
97, 8eqtr4di 2785 . . . . . 6 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
10 fveq2 6900 . . . . . . 7 (𝑘 = 𝐾 → (1.‘𝑘) = (1.‘𝐾))
11 lhpset.u . . . . . . 7 1 = (1.‘𝐾)
1210, 11eqtr4di 2785 . . . . . 6 (𝑘 = 𝐾 → (1.‘𝑘) = 1 )
136, 9, 12breq123d 5164 . . . . 5 (𝑘 = 𝐾 → (𝑤( ⋖ ‘𝑘)(1.‘𝑘) ↔ 𝑤𝐶 1 ))
145, 13rabeqbidv 3446 . . . 4 (𝑘 = 𝐾 → {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)} = {𝑤𝐵𝑤𝐶 1 })
15 df-lhyp 39465 . . . 4 LHyp = (𝑘 ∈ V ↦ {𝑤 ∈ (Base‘𝑘) ∣ 𝑤( ⋖ ‘𝑘)(1.‘𝑘)})
164fvexi 6914 . . . . 5 𝐵 ∈ V
1716rabex 5336 . . . 4 {𝑤𝐵𝑤𝐶 1 } ∈ V
1814, 15, 17fvmpt 7008 . . 3 (𝐾 ∈ V → (LHyp‘𝐾) = {𝑤𝐵𝑤𝐶 1 })
192, 18eqtrid 2779 . 2 (𝐾 ∈ V → 𝐻 = {𝑤𝐵𝑤𝐶 1 })
201, 19syl 17 1 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3428  Vcvv 3471   class class class wbr 5150  cfv 6551  Basecbs 17185  1.cp1 18421  ccvr 38738  LHypclh 39461
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 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-lhyp 39465
This theorem is referenced by:  islhp  39473
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