Nuprl Lemma : geo-not-lt-to-le
∀g:EuclideanPlane. ∀a,b,c,d,e,f:Point. ((¬|ef| < |ab| + |cd|) ⇒ |ab| + |cd| ≤ |ef|)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-le: p ≤ q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
geo-le_wf,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
not-geo-lt,
not_wf,
geo-lt_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
setElimination,
rename,
because_Cache,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
productElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f:Point. ((\mneg{}|ef| < |ab| + |cd|) {}\mRightarrow{} |ab| + |cd| \mleq{} |ef|)
Date html generated:
2019_10_16-PM-02_52_46
Last ObjectModification:
2018_10_04-AM-11_28_08
Theory : euclidean!plane!geometry
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