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Theorem llnset 37071
 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 3429 . 2 (𝐾𝐷𝐾 ∈ V)
2 llnset.n . . 3 𝑁 = (LLines‘𝐾)
3 fveq2 6656 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 llnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2812 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6656 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7 llnset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7eqtr4di 2812 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9 fveq2 6656 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 llnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10eqtr4di 2812 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 5041 . . . . . 6 (𝑘 = 𝐾 → (𝑝( ⋖ ‘𝑘)𝑥𝑝𝐶𝑥))
138, 12rexeqbidv 3321 . . . . 5 (𝑘 = 𝐾 → (∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥 ↔ ∃𝑝𝐴 𝑝𝐶𝑥))
145, 13rabeqbidv 3399 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
15 df-llines 37064 . . . 4 LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
164fvexi 6670 . . . . 5 𝐵 ∈ V
1716rabex 5200 . . . 4 {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥} ∈ V
1814, 15, 17fvmpt 6757 . . 3 (𝐾 ∈ V → (LLines‘𝐾) = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
192, 18syl5eq 2806 . 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 1539   ∈ wcel 2112  ∃wrex 3072  {crab 3075  Vcvv 3410   class class class wbr 5030  ‘cfv 6333  Basecbs 16531   ⋖ ccvr 36828  Atomscatm 36829  LLinesclln 37057 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-iota 6292  df-fun 6335  df-fv 6341  df-llines 37064 This theorem is referenced by:  islln  37072
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