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Theorem llnset 36511
Description: The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnset.b 𝐵 = (Base‘𝐾)
llnset.c 𝐶 = ( ⋖ ‘𝐾)
llnset.a 𝐴 = (Atoms‘𝐾)
llnset.n 𝑁 = (LLines‘𝐾)
Assertion
Ref Expression
llnset (𝐾𝐷𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑥,𝑝,𝐾
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑝)   𝐶(𝑥,𝑝)   𝐷(𝑥,𝑝)   𝑁(𝑥,𝑝)

Proof of Theorem llnset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3517 . 2 (𝐾𝐷𝐾 ∈ V)
2 llnset.n . . 3 𝑁 = (LLines‘𝐾)
3 fveq2 6666 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 llnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2878 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6666 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 llnset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7syl6eqr 2878 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6666 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 llnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10syl6eqr 2878 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 5073 . . . . . 6 (𝑘 = 𝐾 → (𝑝( ⋖ ‘𝑘)𝑥𝑝𝐶𝑥))
138, 12rexeqbidv 3407 . . . . 5 (𝑘 = 𝐾 → (∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥 ↔ ∃𝑝𝐴 𝑝𝐶𝑥))
145, 13rabeqbidv 3490 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
15 df-llines 36504 . . . 4 LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
164fvexi 6680 . . . . 5 𝐵 ∈ V
1716rabex 5231 . . . 4 {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥} ∈ V
1814, 15, 17fvmpt 6764 . . 3 (𝐾 ∈ V → (LLines‘𝐾) = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
192, 18syl5eq 2872 . 2 (𝐾 ∈ V → 𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
201, 19syl 17 1 (𝐾𝐷𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  wrex 3143  {crab 3146  Vcvv 3499   class class class wbr 5062  cfv 6351  Basecbs 16476  ccvr 36268  Atomscatm 36269  LLinesclln 36497
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-llines 36504
This theorem is referenced by:  islln  36512
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