Nuprl Lemma : Euclid-Prop19-weak
∀e:EuclideanPlane. ∀a,b,c:Point. (a # bc ⇒ bca < abc ⇒ (¬¬|ab| < |ac|))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
guard: {T},
subtype_rel: A ⊆r B,
or: P ∨ Q,
prop: ℙ,
uimplies: b supposing a,
stable: Stable{P},
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
member: t ∈ T,
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
uiff: uiff(P;Q),
geo-tri: Triangle(a;b;c)
Lemmas referenced :
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-lsep_wf,
geo-lt-angle_wf,
istype-void,
geo-length-type_wf,
equal_wf,
double-negation-hyp-elim,
geo-lt_wf,
not_wf,
stable__not,
geo-mk-seg_wf,
geo-length_wf,
not-lt-or,
lsep-all-sym,
Euclid-Prop18,
geo-lt-angle-symm2,
geo-lt-angle-symm,
lt-angle-not-cong,
geo-lt-angle-trans,
Euclid-Prop5_1,
geo-congruent-iff-length,
lsep-implies-sep,
geo-sep-sym,
lt-angle-not-cong2,
geo-cong-angle-symmetry
Rules used in proof :
instantiate,
applyEquality,
equalityIstype,
universeIsType,
unionIsType,
inhabitedIsType,
functionIsType,
voidElimination,
lambdaEquality_alt,
unionElimination,
independent_functionElimination,
unionEquality,
independent_isectElimination,
hypothesis,
because_Cache,
rename,
setElimination,
sqequalRule,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} bca < abc {}\mRightarrow{} (\mneg{}\mneg{}|ab| < |ac|))
Date html generated:
2019_10_16-PM-02_15_51
Last ObjectModification:
2018_12_20-PM-01_35_08
Theory : euclidean!plane!geometry
Home
Index