Nuprl Lemma : dist-lemma-le
∀g:EuclideanPlane. ∀a,b,c,d,e,f:Point. (D(a;b;c;d;e;f) ⇒ |ef| ≤ |ab| + |cd|)
Proof
Definitions occuring in Statement :
dist: D(a;b;c;d;e;f),
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],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
dist: D(a;b;c;d;e;f),
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
squash: ↓T,
euclidean-plane: EuclideanPlane,
true: True,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
geo-add-length-between-iff,
geo-le-from-be,
geo-le_wf,
geo-length_wf,
geo-mk-seg_wf,
subtype_rel_self,
iff_weakening_equal,
geo-congruent-iff-length,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
isectElimination,
hypothesis,
independent_isectElimination,
setElimination,
rename,
applyEquality,
lambdaEquality_alt,
imageElimination,
because_Cache,
universeIsType,
inhabitedIsType,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
equalityTransitivity,
independent_functionElimination,
hyp_replacement
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f:Point. (D(a;b;c;d;e;f) {}\mRightarrow{} |ef| \mleq{} |ab| + |cd|)
Date html generated:
2019_10_16-PM-02_48_27
Last ObjectModification:
2018_10_02-PM-02_29_06
Theory : euclidean!plane!geometry
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