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Theorem lplnset 39034
Description: The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
lplnset.b 𝐡 = (Baseβ€˜πΎ)
lplnset.c 𝐢 = ( β‹– β€˜πΎ)
lplnset.n 𝑁 = (LLinesβ€˜πΎ)
lplnset.p 𝑃 = (LPlanesβ€˜πΎ)
Assertion
Ref Expression
lplnset (𝐾 ∈ 𝐴 β†’ 𝑃 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯})
Distinct variable groups:   𝑦,𝑁   π‘₯,𝐡   π‘₯,𝑦,𝐾
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(𝑦)   𝐢(π‘₯,𝑦)   𝑃(π‘₯,𝑦)   𝑁(π‘₯)

Proof of Theorem lplnset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝐴 β†’ 𝐾 ∈ V)
2 lplnset.p . . 3 𝑃 = (LPlanesβ€˜πΎ)
3 fveq2 6902 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
4 lplnset.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2786 . . . . 5 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
6 fveq2 6902 . . . . . . 7 (π‘˜ = 𝐾 β†’ (LLinesβ€˜π‘˜) = (LLinesβ€˜πΎ))
7 lplnset.n . . . . . . 7 𝑁 = (LLinesβ€˜πΎ)
86, 7eqtr4di 2786 . . . . . 6 (π‘˜ = 𝐾 β†’ (LLinesβ€˜π‘˜) = 𝑁)
9 fveq2 6902 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ( β‹– β€˜π‘˜) = ( β‹– β€˜πΎ))
10 lplnset.c . . . . . . . 8 𝐢 = ( β‹– β€˜πΎ)
119, 10eqtr4di 2786 . . . . . . 7 (π‘˜ = 𝐾 β†’ ( β‹– β€˜π‘˜) = 𝐢)
1211breqd 5163 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑦( β‹– β€˜π‘˜)π‘₯ ↔ 𝑦𝐢π‘₯))
138, 12rexeqbidv 3341 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘¦ ∈ (LLinesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯ ↔ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯))
145, 13rabeqbidv 3448 . . . 4 (π‘˜ = 𝐾 β†’ {π‘₯ ∈ (Baseβ€˜π‘˜) ∣ βˆƒπ‘¦ ∈ (LLinesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯} = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯})
15 df-lplanes 39004 . . . 4 LPlanes = (π‘˜ ∈ V ↦ {π‘₯ ∈ (Baseβ€˜π‘˜) ∣ βˆƒπ‘¦ ∈ (LLinesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯})
164fvexi 6916 . . . . 5 𝐡 ∈ V
1716rabex 5338 . . . 4 {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯} ∈ V
1814, 15, 17fvmpt 7010 . . 3 (𝐾 ∈ V β†’ (LPlanesβ€˜πΎ) = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯})
192, 18eqtrid 2780 . 2 (𝐾 ∈ V β†’ 𝑃 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯})
201, 19syl 17 1 (𝐾 ∈ 𝐴 β†’ 𝑃 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067  {crab 3430  Vcvv 3473   class class class wbr 5152  β€˜cfv 6553  Basecbs 17187   β‹– ccvr 38766  LLinesclln 38996  LPlanesclpl 38997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-lplanes 39004
This theorem is referenced by:  islpln  39035
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