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Theorem islpln 39003
Description: The predicate "is a lattice plane". (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
islpln (𝐾 ∈ 𝐴 β†’ (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢𝑋)))
Distinct variable groups:   𝑦,𝑁   𝑦,𝐾   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐡(𝑦)   𝐢(𝑦)   𝑃(𝑦)

Proof of Theorem islpln
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 lplnset.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 lplnset.c . . . 4 𝐢 = ( β‹– β€˜πΎ)
3 lplnset.n . . . 4 𝑁 = (LLinesβ€˜πΎ)
4 lplnset.p . . . 4 𝑃 = (LPlanesβ€˜πΎ)
51, 2, 3, 4lplnset 39002 . . 3 (𝐾 ∈ 𝐴 β†’ 𝑃 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯})
65eleq2d 2815 . 2 (𝐾 ∈ 𝐴 β†’ (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯}))
7 breq2 5152 . . . 4 (π‘₯ = 𝑋 β†’ (𝑦𝐢π‘₯ ↔ 𝑦𝐢𝑋))
87rexbidv 3175 . . 3 (π‘₯ = 𝑋 β†’ (βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯ ↔ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢𝑋))
98elrab 3682 . 2 (𝑋 ∈ {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢π‘₯} ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢𝑋))
106, 9bitrdi 287 1 (𝐾 ∈ 𝐴 β†’ (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘¦ ∈ 𝑁 𝑦𝐢𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3067  {crab 3429   class class class wbr 5148  β€˜cfv 6548  Basecbs 17180   β‹– ccvr 38734  LLinesclln 38964  LPlanesclpl 38965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-lplanes 38972
This theorem is referenced by:  islpln4  39004  lplni  39005  lplnbase  39007  lplnnle2at  39014
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