Nuprl Lemma : lt-angle-irrefl
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point. (abc < xyz ⇒ (¬xyz < abc))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
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,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
or: P ∨ Q,
stable: Stable{P},
iff: P ⇐⇒ Q,
and: P ∧ Q,
basic-geometry-: BasicGeometry-,
geo-lt-angle: abc < xyz,
exists: ∃x:A. B[x],
geo-cong-angle: abc ≅a xyz,
geo-eq: a ≡ b,
oriented-plane: OrientedPlane,
rev_implies: P ⇐ Q
Lemmas referenced :
geo-lt-angle_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
stable__false,
false_wf,
geo-lsep_wf,
not_wf,
istype-void,
minimal-double-negation-hyp-elim,
not-lsep-iff-colinear,
minimal-not-not-excluded-middle,
lt-angle-not-cong,
geo-lt-angle-trans,
geo-colinear-cases,
geo-eq_wf,
geo-strict-between_wf,
not-lt-zero-angle,
geo-between_wf,
geo-between-trivial,
geo-colinear_functionality,
geo-eq_weakening,
geo-lsep_functionality,
geo-lt-angle_functionality,
geo-between_functionality,
straight-angles-not-lt2,
geo-strict-between-implies-between,
geo-between-symmetry,
zero-angles-congruent,
geo-sep-sym,
lt-angle-not-cong2,
straight-angles-not-lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
universeIsType,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
inhabitedIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
unionEquality,
functionEquality,
unionIsType,
functionIsType,
unionElimination,
productElimination,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (abc < xyz {}\mRightarrow{} (\mneg{}xyz < abc))
Date html generated:
2019_10_16-PM-02_02_27
Last ObjectModification:
2019_09_12-AM-11_41_08
Theory : euclidean!plane!geometry
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