Nuprl Lemma : Euclid-drop-perp-00
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . ∀c:Point. ∃x,p:Point. (Colinear(p;x;c) ∧ ab ⊥p px ∧ x # ab ∧ x ≠ c)
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
Euclid-drop-perp-0-ext,
sq_stable__colinear,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
sq_stable__geo-perp-in,
sq_stable__geo-lsep,
sq_stable__geo-sep,
geo-colinear_wf,
geo-perp-in_wf,
geo-lsep_wf,
geo-sep_wf,
exists_wf,
geo-point_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation,
setElimination,
rename,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
productEquality,
lambdaEquality,
setEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . \mforall{}c:Point.
\mexists{}x,p:Point. (Colinear(p;x;c) \mwedge{} ab \mbot{}p px \mwedge{} x \# ab \mwedge{} x \mneq{} c)
Date html generated:
2018_05_22-PM-00_11_29
Last ObjectModification:
2018_05_19-PM-08_48_45
Theory : euclidean!plane!geometry
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