Nuprl Lemma : geo-congruent-mid-exists
∀e:HeytingGeometry. ∀A,B,C:Point. (A # BC ⇒ CA ≅ CB ⇒ (∃x:Point. A=x=B))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-midpoint: a=m=b,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
heyting-geometry: Error :heyting-geometry,
prop: ℙ,
uimplies: b supposing a,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
cand: A c∧ B,
and: P ∧ Q,
guard: {T},
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-point_wf,
Error :geo-triangle_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
Error :heyting-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-length-flip,
heyting-geometry-subtype,
geo-congruent-iff-length,
geo-triangle-symmetry,
isosceles-mid-exists
Rules used in proof :
rename,
setElimination,
instantiate,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
isectElimination,
sqequalRule,
applyEquality,
productElimination,
hypothesis,
because_Cache,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:HeytingGeometry. \mforall{}A,B,C:Point. (A \# BC {}\mRightarrow{} CA \00D0 CB {}\mRightarrow{} (\mexists{}x:Point. A=x=B))
Date html generated:
2017_10_02-PM-07_09_30
Last ObjectModification:
2017_08_08-PM-00_41_08
Theory : euclidean!plane!geometry
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