Nuprl Lemma : geo-le-congruent
∀g:BasicGeometry. ∀a,b,c,d:Point. ((|ab| ≤ |cd| ∧ |cd| ≤ |ab|) ⇒ ab ≅ cd)
Proof
Definitions occuring in Statement :
geo-le: p ≤ q,
geo-length: |s|,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
uimplies: b supposing a,
uiff: uiff(P;Q),
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
geo-le_antisymmetry,
geo-length_wf,
geo-mk-seg_wf,
geo-congruent-iff-length,
geo-le_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
setElimination,
rename,
hypothesis,
because_Cache,
independent_isectElimination,
equalitySymmetry,
sqequalRule,
productIsType,
universeIsType,
inhabitedIsType,
applyEquality,
instantiate
Latex:
\mforall{}g:BasicGeometry. \mforall{}a,b,c,d:Point. ((|ab| \mleq{} |cd| \mwedge{} |cd| \mleq{} |ab|) {}\mRightarrow{} ab \mcong{} cd)
Date html generated:
2019_10_16-PM-01_36_13
Last ObjectModification:
2018_10_03-PM-00_05_03
Theory : euclidean!plane!geometry
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