Nuprl Lemma : geo-midpoint_functionality
∀e:BasicGeometry. ∀m,a,b,m',a',b':Point. (m ≡ m' ⇒ a ≡ a' ⇒ b ≡ b' ⇒ {a=m=b ⇒ a'=m'=b'})
Proof
Definitions occuring in Statement :
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T},
geo-midpoint: a=m=b,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-point_wf,
geo-eq_wf,
geo-congruent_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-between_wf,
geo-between_functionality,
geo-eq_weakening,
geo-congruent_functionality
Rules used in proof :
because_Cache,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
productEquality,
hypothesis,
independent_pairFormation,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}m,a,b,m',a',b':Point. (m \mequiv{} m' {}\mRightarrow{} a \mequiv{} a' {}\mRightarrow{} b \mequiv{} b' {}\mRightarrow{} \{a=m=b {}\mRightarrow{} a'=m'=b'\})
Date html generated:
2017_10_02-PM-04_45_45
Last ObjectModification:
2017_08_05-AM-11_45_50
Theory : euclidean!plane!geometry
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