Nuprl Lemma : geo-le-sep
∀e:BasicGeometry. ∀A,B,C,P:Point. (|AB| ≤ |CP| ⇒ A ≠ B ⇒ C ≠ P)
Proof
Definitions occuring in Statement :
geo-le: p ≤ q,
geo-length: |s|,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
basic-geometry: BasicGeometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-point_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-le_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-le-iff,
geo-ge-sep
Rules used in proof :
because_Cache,
rename,
setElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,C,P:Point. (|AB| \mleq{} |CP| {}\mRightarrow{} A \mneq{} B {}\mRightarrow{} C \mneq{} P)
Date html generated:
2017_10_02-PM-06_18_19
Last ObjectModification:
2017_08_05-PM-04_13_05
Theory : euclidean!plane!geometry
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