Nuprl Lemma : dist-lemma-lt-2
∀g:EuclideanPlane. ∀a,b,e,f:Point. (D(a;b;b;b;e;f) ⇒ |ef| < |ab|)
Proof
Definitions occuring in Statement :
dist: D(a;b;c;d;e;f),
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
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,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry
Lemmas referenced :
dist_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-add-length-zero2,
geo-length_wf,
geo-mk-seg_wf,
geo-lt_wf,
dist-lemma-lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,e,f:Point. (D(a;b;b;b;e;f) {}\mRightarrow{} |ef| < |ab|)
Date html generated:
2019_10_16-PM-02_49_04
Last ObjectModification:
2019_06_05-PM-01_40_31
Theory : euclidean!plane!geometry
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