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Theorem lplnset 36217
Description: The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
lplnset.b 𝐵 = (Base‘𝐾)
lplnset.c 𝐶 = ( ⋖ ‘𝐾)
lplnset.n 𝑁 = (LLines‘𝐾)
lplnset.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
lplnset (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
Distinct variable groups:   𝑦,𝑁   𝑥,𝐵   𝑥,𝑦,𝐾
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝑁(𝑥)

Proof of Theorem lplnset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3458 . 2 (𝐾𝐴𝐾 ∈ V)
2 lplnset.p . . 3 𝑃 = (LPlanes‘𝐾)
3 fveq2 6545 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4 lplnset.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2851 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6 fveq2 6545 . . . . . . 7 (𝑘 = 𝐾 → (LLines‘𝑘) = (LLines‘𝐾))
7 lplnset.n . . . . . . 7 𝑁 = (LLines‘𝐾)
86, 7syl6eqr 2851 . . . . . 6 (𝑘 = 𝐾 → (LLines‘𝑘) = 𝑁)
9 fveq2 6545 . . . . . . . 8 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = ( ⋖ ‘𝐾))
10 lplnset.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
119, 10syl6eqr 2851 . . . . . . 7 (𝑘 = 𝐾 → ( ⋖ ‘𝑘) = 𝐶)
1211breqd 4979 . . . . . 6 (𝑘 = 𝐾 → (𝑦( ⋖ ‘𝑘)𝑥𝑦𝐶𝑥))
138, 12rexeqbidv 3364 . . . . 5 (𝑘 = 𝐾 → (∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑥))
145, 13rabeqbidv 3433 . . . 4 (𝑘 = 𝐾 → {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥} = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
15 df-lplanes 36187 . . . 4 LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑦 ∈ (LLines‘𝑘)𝑦( ⋖ ‘𝑘)𝑥})
164fvexi 6559 . . . . 5 𝐵 ∈ V
1716rabex 5133 . . . 4 {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ∈ V
1814, 15, 17fvmpt 6642 . . 3 (𝐾 ∈ V → (LPlanes‘𝐾) = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
192, 18syl5eq 2845 . 2 (𝐾 ∈ V → 𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
201, 19syl 17 1 (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1525  wcel 2083  wrex 3108  {crab 3111  Vcvv 3440   class class class wbr 4968  cfv 6232  Basecbs 16316  ccvr 35950  LLinesclln 36179  LPlanesclpl 36180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-lplanes 36187
This theorem is referenced by:  islpln  36218
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