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Theorem llnset 35312
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 3352 . 2 (𝐾𝐷𝐾 ∈ V)
2 llnset.n . . 3 𝑁 = (LLines‘𝐾)
3 fveq2 6353 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 llnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2812 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6353 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 llnset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7syl6eqr 2812 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6353 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 llnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10syl6eqr 2812 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 4815 . . . . . 6 (𝑘 = 𝐾 → (𝑝( ⋖ ‘𝑘)𝑥𝑝𝐶𝑥))
138, 12rexeqbidv 3292 . . . . 5 (𝑘 = 𝐾 → (∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥 ↔ ∃𝑝𝐴 𝑝𝐶𝑥))
145, 13rabeqbidv 3335 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
15 df-llines 35305 . . . 4 LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
16 fvex 6363 . . . . . 6 (Base‘𝐾) ∈ V
174, 16eqeltri 2835 . . . . 5 𝐵 ∈ V
1817rabex 4964 . . . 4 {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥} ∈ V
1914, 15, 18fvmpt 6445 . . 3 (𝐾 ∈ V → (LLines‘𝐾) = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
202, 19syl5eq 2806 . 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 1632  wcel 2139  wrex 3051  {crab 3054  Vcvv 3340   class class class wbr 4804  cfv 6049  Basecbs 16079  ccvr 35070  Atomscatm 35071  LLinesclln 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-llines 35305
This theorem is referenced by:  islln  35313
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