Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islpln Structured version   Visualization version   GIF version

Theorem islpln 36702
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 36701 . . 3 (𝐾𝐴𝑃 = {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥})
65eleq2d 2896 . 2 (𝐾𝐴 → (𝑋𝑃𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥}))
7 breq2 5046 . . . 4 (𝑥 = 𝑋 → (𝑦𝐶𝑥𝑦𝐶𝑋))
87rexbidv 3284 . . 3 (𝑥 = 𝑋 → (∃𝑦𝑁 𝑦𝐶𝑥 ↔ ∃𝑦𝑁 𝑦𝐶𝑋))
98elrab 3660 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑁 𝑦𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋))
106, 9syl6bb 289 1 (𝐾𝐴 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wrex 3126  {crab 3129   class class class wbr 5042  cfv 6331  Basecbs 16462  ccvr 36434  LLinesclln 36663  LPlanesclpl 36664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-lplanes 36671
This theorem is referenced by:  islpln4  36703  lplni  36704  lplnbase  36706  lplnnle2at  36713
  Copyright terms: Public domain W3C validator