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Theorem llnset 38364
Description: The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnset.b 𝐡 = (Baseβ€˜πΎ)
llnset.c 𝐢 = ( β‹– β€˜πΎ)
llnset.a 𝐴 = (Atomsβ€˜πΎ)
llnset.n 𝑁 = (LLinesβ€˜πΎ)
Assertion
Ref Expression
llnset (𝐾 ∈ 𝐷 β†’ 𝑁 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯})
Distinct variable groups:   𝐴,𝑝   π‘₯,𝐡   π‘₯,𝑝,𝐾
Allowed substitution hints:   𝐴(π‘₯)   𝐡(𝑝)   𝐢(π‘₯,𝑝)   𝐷(π‘₯,𝑝)   𝑁(π‘₯,𝑝)

Proof of Theorem llnset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝐷 β†’ 𝐾 ∈ V)
2 llnset.n . . 3 𝑁 = (LLinesβ€˜πΎ)
3 fveq2 6888 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
4 llnset.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2790 . . . . 5 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
6 fveq2 6888 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
7 llnset.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
86, 7eqtr4di 2790 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
9 fveq2 6888 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ( β‹– β€˜π‘˜) = ( β‹– β€˜πΎ))
10 llnset.c . . . . . . . 8 𝐢 = ( β‹– β€˜πΎ)
119, 10eqtr4di 2790 . . . . . . 7 (π‘˜ = 𝐾 β†’ ( β‹– β€˜π‘˜) = 𝐢)
1211breqd 5158 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑝( β‹– β€˜π‘˜)π‘₯ ↔ 𝑝𝐢π‘₯))
138, 12rexeqbidv 3343 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)𝑝( β‹– β€˜π‘˜)π‘₯ ↔ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯))
145, 13rabeqbidv 3449 . . . 4 (π‘˜ = 𝐾 β†’ {π‘₯ ∈ (Baseβ€˜π‘˜) ∣ βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)𝑝( β‹– β€˜π‘˜)π‘₯} = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯})
15 df-llines 38357 . . . 4 LLines = (π‘˜ ∈ V ↦ {π‘₯ ∈ (Baseβ€˜π‘˜) ∣ βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)𝑝( β‹– β€˜π‘˜)π‘₯})
164fvexi 6902 . . . . 5 𝐡 ∈ V
1716rabex 5331 . . . 4 {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯} ∈ V
1814, 15, 17fvmpt 6995 . . 3 (𝐾 ∈ V β†’ (LLinesβ€˜πΎ) = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯})
192, 18eqtrid 2784 . 2 (𝐾 ∈ V β†’ 𝑁 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯})
201, 19syl 17 1 (𝐾 ∈ 𝐷 β†’ 𝑁 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  Vcvv 3474   class class class wbr 5147  β€˜cfv 6540  Basecbs 17140   β‹– ccvr 38120  Atomscatm 38121  LLinesclln 38350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-llines 38357
This theorem is referenced by:  islln  38365
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