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Theorem lineset 34525
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 3198 . 2 (𝐾𝐵𝐾 ∈ V)
2 lineset.n . . 3 𝑁 = (Lines‘𝐾)
3 fveq2 6150 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 lineset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2673 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6150 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
7 lineset.l . . . . . . . . . . . . 13 = (le‘𝐾)
86, 7syl6eqr 2673 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = )
98breqd 4626 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 (𝑞(join‘𝑘)𝑟)))
10 fveq2 6150 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
11 lineset.j . . . . . . . . . . . . . 14 = (join‘𝐾)
1210, 11syl6eqr 2673 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = )
1312oveqd 6624 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (𝑞(join‘𝑘)𝑟) = (𝑞 𝑟))
1413breq2d 4627 . . . . . . . . . . 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 3181 . . . . . . . . 9 (𝑘 = 𝐾 → {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} = {𝑝𝐴𝑝 (𝑞 𝑟)})
1716eqeq2d 2631 . . . . . . . 8 (𝑘 = 𝐾 → (𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} ↔ 𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
1817anbi2d 739 . . . . . . 7 (𝑘 = 𝐾 → ((𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
195, 18rexeqbidv 3142 . . . . . 6 (𝑘 = 𝐾 → (∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ ∃𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
205, 19rexeqbidv 3142 . . . . 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 2738 . . . 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 34288 . . . 4 Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
23 fvex 6160 . . . . . 6 (Atoms‘𝐾) ∈ V
244, 23eqeltri 2694 . . . . 5 𝐴 ∈ V
25 df-sn 4151 . . . . . . 7 {{𝑝𝐴𝑝 (𝑞 𝑟)}} = {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}}
26 snex 4871 . . . . . . 7 {{𝑝𝐴𝑝 (𝑞 𝑟)}} ∈ V
2725, 26eqeltrri 2695 . . . . . 6 {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}} ∈ V
28 simpr 477 . . . . . . 7 ((𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})
2928ss2abi 3655 . . . . . 6 {𝑠 ∣ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ⊆ {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}}
3027, 29ssexi 4765 . . . . 5 {𝑠 ∣ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ∈ V
3124, 24, 30ab2rexex2 7108 . . . 4 {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ∈ V
3221, 22, 31fvmpt 6241 . . 3 (𝐾 ∈ V → (Lines‘𝐾) = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
332, 32syl5eq 2667 . 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 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wrex 2908  {crab 2911  Vcvv 3186  {csn 4150   class class class wbr 4615  cfv 5849  (class class class)co 6607  lecple 15872  joincjn 16868  Atomscatm 34051  Linesclines 34281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-lines 34288
This theorem is referenced by:  isline  34526
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