Nuprl Lemma : geo-congruent-strictbetween-exists
∀e:BasicGeometry. ∀a,b,c,a',c':Point. (a-b-c ⇒ ac ≅ a'c' ⇒ (∃b':Point. ((a'-b'-c' ∧ ab ≅ a'b') ∧ bc ≅ b'c')))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
geo-strict-between: a-b-c
Lemmas referenced :
geo-point_wf,
geo-congruent_wf,
geo-strict-between_wf,
geo-strict-between-implies-between,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
basic-geometry--subtype,
geo-strict-between-sep2,
geo-congruent-between-exists,
geo-congruent-sep,
geo-congruent-symmetry
Rules used in proof :
productEquality,
independent_pairFormation,
dependent_pairFormation,
productElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
isectElimination,
instantiate,
applyEquality,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',c':Point.
(a-b-c {}\mRightarrow{} ac \00D0 a'c' {}\mRightarrow{} (\mexists{}b':Point. ((a'-b'-c' \mwedge{} ab \00D0 a'b') \mwedge{} bc \00D0 b'c')))
Date html generated:
2017_10_02-PM-06_15_18
Last ObjectModification:
2017_08_05-PM-04_12_01
Theory : euclidean!plane!geometry
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