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Theorem lplnset 38021
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 3466 . 2 (𝐾 ∈ 𝐴 β†’ 𝐾 ∈ V)
2 lplnset.p . . 3 𝑃 = (LPlanesβ€˜πΎ)
3 fveq2 6847 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
4 lplnset.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2795 . . . . 5 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
6 fveq2 6847 . . . . . . 7 (π‘˜ = 𝐾 β†’ (LLinesβ€˜π‘˜) = (LLinesβ€˜πΎ))
7 lplnset.n . . . . . . 7 𝑁 = (LLinesβ€˜πΎ)
86, 7eqtr4di 2795 . . . . . 6 (π‘˜ = 𝐾 β†’ (LLinesβ€˜π‘˜) = 𝑁)
9 fveq2 6847 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ( β‹– β€˜π‘˜) = ( β‹– β€˜πΎ))
10 lplnset.c . . . . . . . 8 𝐢 = ( β‹– β€˜πΎ)
119, 10eqtr4di 2795 . . . . . . 7 (π‘˜ = 𝐾 β†’ ( β‹– β€˜π‘˜) = 𝐢)
1211breqd 5121 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑦( β‹– β€˜π‘˜)π‘₯ ↔ 𝑦𝐢π‘₯))
138, 12rexeqbidv 3323 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘¦ ∈ (LLinesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯ ↔ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯))
145, 13rabeqbidv 3427 . . . 4 (π‘˜ = 𝐾 β†’ {π‘₯ ∈ (Baseβ€˜π‘˜) ∣ βˆƒπ‘¦ ∈ (LLinesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯} = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯})
15 df-lplanes 37991 . . . 4 LPlanes = (π‘˜ ∈ V ↦ {π‘₯ ∈ (Baseβ€˜π‘˜) ∣ βˆƒπ‘¦ ∈ (LLinesβ€˜π‘˜)𝑦( β‹– β€˜π‘˜)π‘₯})
164fvexi 6861 . . . . 5 𝐡 ∈ V
1716rabex 5294 . . . 4 {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯} ∈ V
1814, 15, 17fvmpt 6953 . . 3 (𝐾 ∈ V β†’ (LPlanesβ€˜πΎ) = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯})
192, 18eqtrid 2789 . 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 1542   ∈ wcel 2107  βˆƒwrex 3074  {crab 3410  Vcvv 3448   class class class wbr 5110  β€˜cfv 6501  Basecbs 17090   β‹– ccvr 37753  LLinesclln 37983  LPlanesclpl 37984
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-lplanes 37991
This theorem is referenced by:  islpln  38022
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