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Theorem islpln 35542
Description: The predicate "is a lattice plane". (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
islpln (𝐾𝐴 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋)))
Distinct variable groups:   𝑦,𝑁   𝑦,𝐾   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝑃(𝑦)

Proof of Theorem islpln
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 lplnset.b . . . 4 𝐵 = (Base‘𝐾)
2 lplnset.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
3 lplnset.n . . . 4 𝑁 = (LLines‘𝐾)
4 lplnset.p . . . 4 𝑃 = (LPlanes‘𝐾)
51, 2, 3, 4lplnset 35541 . . 3 (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
65eleq2d 2862 . 2 (𝐾𝐴 → (𝑋𝑃𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥}))
7 breq2 4845 . . . 4 (𝑥 = 𝑋 → (𝑦𝐶𝑥𝑦𝐶𝑋))
87rexbidv 3231 . . 3 (𝑥 = 𝑋 → (∃𝑦𝑁 𝑦𝐶𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑋))
98elrab 3554 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋))
106, 9syl6bb 279 1 (𝐾𝐴 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3088  {crab 3091   class class class wbr 4841  cfv 6099  Basecbs 16180  ccvr 35274  LLinesclln 35503  LPlanesclpl 35504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107  df-lplanes 35511
This theorem is referenced by:  islpln4  35543  lplni  35544  lplnbase  35546  lplnnle2at  35553
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