Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  llnset Structured version   Visualization version   GIF version

Theorem llnset 33612
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 3184 . 2 (𝐾𝐷𝐾 ∈ V)
2 llnset.n . . 3 𝑁 = (LLines‘𝐾)
3 fveq2 6088 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 llnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2661 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6088 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 llnset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7syl6eqr 2661 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6088 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 llnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10syl6eqr 2661 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 4588 . . . . . 6 (𝑘 = 𝐾 → (𝑝( ⋖ ‘𝑘)𝑥𝑝𝐶𝑥))
138, 12rexeqbidv 3129 . . . . 5 (𝑘 = 𝐾 → (∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥 ↔ ∃𝑝𝐴 𝑝𝐶𝑥))
145, 13rabeqbidv 3167 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
15 df-llines 33605 . . . 4 LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
16 fvex 6098 . . . . . 6 (Base‘𝐾) ∈ V
174, 16eqeltri 2683 . . . . 5 𝐵 ∈ V
1817rabex 4735 . . . 4 {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥} ∈ V
1914, 15, 18fvmpt 6176 . . 3 (𝐾 ∈ V → (LLines‘𝐾) = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
202, 19syl5eq 2655 . 2 (𝐾 ∈ V → 𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
211, 20syl 17 1 (𝐾𝐷𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wrex 2896  {crab 2899  Vcvv 3172   class class class wbr 4577  cfv 5790  Basecbs 15641  ccvr 33370  Atomscatm 33371  LLinesclln 33598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-llines 33605
This theorem is referenced by:  islln  33613
  Copyright terms: Public domain W3C validator