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Theorem lineset 35626
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 3364 . 2 (𝐾𝐵𝐾 ∈ V)
2 lineset.n . . 3 𝑁 = (Lines‘𝐾)
3 fveq2 6374 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 lineset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2816 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6374 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
7 lineset.l . . . . . . . . . . . . 13 = (le‘𝐾)
86, 7syl6eqr 2816 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (le‘𝑘) = )
98breqd 4819 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 (𝑞(join‘𝑘)𝑟)))
10 fveq2 6374 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
11 lineset.j . . . . . . . . . . . . . 14 = (join‘𝐾)
1210, 11syl6eqr 2816 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = )
1312oveqd 6858 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (𝑞(join‘𝑘)𝑟) = (𝑞 𝑟))
1413breq2d 4820 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑝 (𝑞(join‘𝑘)𝑟) ↔ 𝑝 (𝑞 𝑟)))
159, 14bitrd 270 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 (𝑞 𝑟)))
165, 15rabeqbidv 3343 . . . . . . . . 9 (𝑘 = 𝐾 → {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} = {𝑝𝐴𝑝 (𝑞 𝑟)})
1716eqeq2d 2774 . . . . . . . 8 (𝑘 = 𝐾 → (𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} ↔ 𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
1817anbi2d 622 . . . . . . 7 (𝑘 = 𝐾 → ((𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
195, 18rexeqbidv 3300 . . . . . 6 (𝑘 = 𝐾 → (∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ ∃𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
205, 19rexeqbidv 3300 . . . . 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 2883 . . . 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 35389 . . . 4 Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
234fvexi 6388 . . . . 5 𝐴 ∈ V
24 df-sn 4334 . . . . . . 7 {{𝑝𝐴𝑝 (𝑞 𝑟)}} = {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}}
25 snex 5063 . . . . . . 7 {{𝑝𝐴𝑝 (𝑞 𝑟)}} ∈ V
2624, 25eqeltrri 2840 . . . . . 6 {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}} ∈ V
27 simpr 477 . . . . . . 7 ((𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})
2827ss2abi 3833 . . . . . 6 {𝑠 ∣ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ⊆ {𝑠𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)}}
2926, 28ssexi 4963 . . . . 5 {𝑠 ∣ (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ∈ V
3023, 23, 29ab2rexex2 7357 . . . 4 {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ∈ V
3121, 22, 30fvmpt 6470 . . 3 (𝐾 ∈ V → (Lines‘𝐾) = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
322, 31syl5eq 2810 . 2 (𝐾 ∈ V → 𝑁 = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
331, 32syl 17 1 (𝐾𝐵𝑁 = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {cab 2750  wne 2936  wrex 3055  {crab 3058  Vcvv 3349  {csn 4333   class class class wbr 4808  cfv 6067  (class class class)co 6841  lecple 16222  joincjn 17211  Atomscatm 35151  Linesclines 35382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-ov 6844  df-lines 35389
This theorem is referenced by:  isline  35627
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