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Theorem islpln4 36785
Description: The predicate "is a lattice plane". (Contributed by NM, 17-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
islpln4 ((𝐾𝐴𝑋𝐵) → (𝑋𝑃 ↔ ∃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 islpln4
StepHypRef Expression
1 lplnset.b . . 3 𝐵 = (Base‘𝐾)
2 lplnset.c . . 3 𝐶 = ( ⋖ ‘𝐾)
3 lplnset.n . . 3 𝑁 = (LLines‘𝐾)
4 lplnset.p . . 3 𝑃 = (LPlanes‘𝐾)
51, 2, 3, 4islpln 36784 . 2 (𝐾𝐴 → (𝑋𝑃 ↔ (𝑋𝐵 ∧ ∃𝑦𝑁 𝑦𝐶𝑋)))
65baibd 543 1 ((𝐾𝐴𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑦𝑁 𝑦𝐶𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wrex 3131   class class class wbr 5042  cfv 6334  Basecbs 16474  ccvr 36516  LLinesclln 36745  LPlanesclpl 36746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-lplanes 36753
This theorem is referenced by:  islpln3  36787  lplncmp  36816
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