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Theorem lineset 38609
Description: The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)
Hypotheses
Ref Expression
lineset.l ≀ = (leβ€˜πΎ)
lineset.j ∨ = (joinβ€˜πΎ)
lineset.a 𝐴 = (Atomsβ€˜πΎ)
lineset.n 𝑁 = (Linesβ€˜πΎ)
Assertion
Ref Expression
lineset (𝐾 ∈ 𝐡 β†’ 𝑁 = {𝑠 ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})})
Distinct variable groups:   π‘ž,𝑝,π‘Ÿ,𝑠,𝐴   𝐾,𝑝,π‘ž,π‘Ÿ,𝑠   ∨ ,𝑠   ≀ ,𝑠
Allowed substitution hints:   𝐡(𝑠,π‘Ÿ,π‘ž,𝑝)   ∨ (π‘Ÿ,π‘ž,𝑝)   ≀ (π‘Ÿ,π‘ž,𝑝)   𝑁(𝑠,π‘Ÿ,π‘ž,𝑝)

Proof of Theorem lineset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3493 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 lineset.n . . 3 𝑁 = (Linesβ€˜πΎ)
3 fveq2 6892 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 lineset.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2791 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
6 fveq2 6892 . . . . . . . . . . . . 13 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
7 lineset.l . . . . . . . . . . . . 13 ≀ = (leβ€˜πΎ)
86, 7eqtr4di 2791 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
98breqd 5160 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ) ↔ 𝑝 ≀ (π‘ž(joinβ€˜π‘˜)π‘Ÿ)))
10 fveq2 6892 . . . . . . . . . . . . . 14 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = (joinβ€˜πΎ))
11 lineset.j . . . . . . . . . . . . . 14 ∨ = (joinβ€˜πΎ)
1210, 11eqtr4di 2791 . . . . . . . . . . . . 13 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = ∨ )
1312oveqd 7426 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ (π‘ž(joinβ€˜π‘˜)π‘Ÿ) = (π‘ž ∨ π‘Ÿ))
1413breq2d 5161 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (𝑝 ≀ (π‘ž(joinβ€˜π‘˜)π‘Ÿ) ↔ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)))
159, 14bitrd 279 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ) ↔ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)))
165, 15rabeqbidv 3450 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ {𝑝 ∈ (Atomsβ€˜π‘˜) ∣ 𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})
1716eqeq2d 2744 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (𝑠 = {𝑝 ∈ (Atomsβ€˜π‘˜) ∣ 𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ)} ↔ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}))
1817anbi2d 630 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ (Atomsβ€˜π‘˜) ∣ 𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ)}) ↔ (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
195, 18rexeqbidv 3344 . . . . . 6 (π‘˜ = 𝐾 β†’ (βˆƒπ‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ (Atomsβ€˜π‘˜) ∣ 𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ)}) ↔ βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
205, 19rexeqbidv 3344 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘ž ∈ (Atomsβ€˜π‘˜)βˆƒπ‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ (Atomsβ€˜π‘˜) ∣ 𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ)}) ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
2120abbidv 2802 . . . 4 (π‘˜ = 𝐾 β†’ {𝑠 ∣ βˆƒπ‘ž ∈ (Atomsβ€˜π‘˜)βˆƒπ‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ (Atomsβ€˜π‘˜) ∣ 𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ)})} = {𝑠 ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})})
22 df-lines 38372 . . . 4 Lines = (π‘˜ ∈ V ↦ {𝑠 ∣ βˆƒπ‘ž ∈ (Atomsβ€˜π‘˜)βˆƒπ‘Ÿ ∈ (Atomsβ€˜π‘˜)(π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ (Atomsβ€˜π‘˜) ∣ 𝑝(leβ€˜π‘˜)(π‘ž(joinβ€˜π‘˜)π‘Ÿ)})})
234fvexi 6906 . . . . 5 𝐴 ∈ V
24 df-sn 4630 . . . . . . 7 {{𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}} = {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}}
25 snex 5432 . . . . . . 7 {{𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}} ∈ V
2624, 25eqeltrri 2831 . . . . . 6 {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}} ∈ V
27 simpr 486 . . . . . . 7 ((π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}) β†’ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})
2827ss2abi 4064 . . . . . 6 {𝑠 ∣ (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})} βŠ† {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}}
2926, 28ssexi 5323 . . . . 5 {𝑠 ∣ (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})} ∈ V
3023, 23, 29ab2rexex2 7967 . . . 4 {𝑠 ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})} ∈ V
3121, 22, 30fvmpt 6999 . . 3 (𝐾 ∈ V β†’ (Linesβ€˜πΎ) = {𝑠 ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})})
322, 31eqtrid 2785 . 2 (𝐾 ∈ V β†’ 𝑁 = {𝑠 ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})})
331, 32syl 17 1 (𝐾 ∈ 𝐡 β†’ 𝑁 = {𝑠 ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆƒwrex 3071  {crab 3433  Vcvv 3475  {csn 4629   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  lecple 17204  joincjn 18264  Atomscatm 38133  Linesclines 38365
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-lines 38372
This theorem is referenced by:  isline  38610
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