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Theorem lineset 35342
Description: The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)
Hypotheses
Ref Expression
lineset.l = (le‘𝐾)
lineset.j = (join‘𝐾)
lineset.a 𝐴 = (Atoms‘𝐾)
lineset.n 𝑁 = (Lines‘𝐾)
Assertion
Ref Expression
lineset (𝐾𝐵𝑁 = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝐴   𝐾,𝑝,𝑞,𝑟,𝑠   ,𝑠   ,𝑠
Allowed substitution hints:   𝐵(𝑠,𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   𝑁(𝑠,𝑟,𝑞,𝑝)

Proof of Theorem lineset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3243 . 2 (𝐾𝐵𝐾 ∈ V)
2 lineset.n . . 3 𝑁 = (Lines‘𝐾)
3 fveq2 6229 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 lineset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2703 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6229 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
7 lineset.l . . . . . . . . . . . . 13 = (le‘𝐾)
86, 7syl6eqr 2703 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = )
98breqd 4696 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 (𝑞(join‘𝑘)𝑟)))
10 fveq2 6229 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
11 lineset.j . . . . . . . . . . . . . 14 = (join‘𝐾)
1210, 11syl6eqr 2703 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = )
1312oveqd 6707 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (𝑞(join‘𝑘)𝑟) = (𝑞 𝑟))
1413breq2d 4697 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝 (𝑞(join‘𝑘)𝑟) ↔ 𝑝 (𝑞 𝑟)))
159, 14bitrd 268 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 (𝑞 𝑟)))
165, 15rabeqbidv 3226 . . . . . . . . 9 (𝑘 = 𝐾 → {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} = {𝑝𝐴𝑝 (𝑞 𝑟)})
1716eqeq2d 2661 . . . . . . . 8 (𝑘 = 𝐾 → (𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} ↔ 𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
1817anbi2d 740 . . . . . . 7 (𝑘 = 𝐾 → ((𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
195, 18rexeqbidv 3183 . . . . . 6 (𝑘 = 𝐾 → (∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ ∃𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
205, 19rexeqbidv 3183 . . . . 5 (𝑘 = 𝐾 → (∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
2120abbidv 2770 . . . 4 (𝑘 = 𝐾 → {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})} = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
22 df-lines 35105 . . . 4 Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
23 fvex 6239 . . . . . 6 (Atoms‘𝐾) ∈ V
244, 23eqeltri 2726 . . . . 5 𝐴 ∈ V
25 df-sn 4211 . . . . . . 7 {{𝑝𝐴𝑝 (𝑞 𝑟)}} = {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}}
26 snex 4938 . . . . . . 7 {{𝑝𝐴𝑝 (𝑞 𝑟)}} ∈ V
2725, 26eqeltrri 2727 . . . . . 6 {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}} ∈ V
28 simpr 476 . . . . . . 7 ((𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})
2928ss2abi 3707 . . . . . 6 {𝑠 ∣ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ⊆ {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}}
3027, 29ssexi 4836 . . . . 5 {𝑠 ∣ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ∈ V
3124, 24, 30ab2rexex2 7202 . . . 4 {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ∈ V
3221, 22, 31fvmpt 6321 . . 3 (𝐾 ∈ V → (Lines‘𝐾) = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
332, 32syl5eq 2697 . 2 (𝐾 ∈ V → 𝑁 = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
341, 33syl 17 1 (𝐾𝐵𝑁 = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wrex 2942  {crab 2945  Vcvv 3231  {csn 4210   class class class wbr 4685  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  Atomscatm 34868  Linesclines 35098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-lines 35105
This theorem is referenced by:  isline  35343
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