Nuprl Lemma : geo-lt-sep
∀e:BasicGeometry. ∀a,b,c,d:Point. (|ab| < |cd| ⇒ c ≠ d)
Proof
Definitions occuring in Statement :
geo-lt: 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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo_ge: geo_ge(e; p; q)
Lemmas referenced :
geo-zero-lt-iff,
geo-zero-le,
geo-length_wf,
geo-mk-seg_wf,
geo-lt_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-zero-length_wf,
geo-lt_functionality_wrt_implies,
geo-le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairFormation,
independent_functionElimination,
isectElimination,
setElimination,
rename,
universeIsType,
because_Cache,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (|ab| < |cd| {}\mRightarrow{} c \mneq{} d)
Date html generated:
2019_10_16-PM-01_19_53
Last ObjectModification:
2018_10_02-PM-02_27_34
Theory : euclidean!plane!geometry
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