Nuprl Lemma : straight-angles-not-lt
∀g:EuclideanPlane. ∀a,b,c,x,y,z:Point. (a-b-c ⇒ x-y-z ⇒ (¬abc < xyz))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
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,
geo-lt-angle: abc < xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
cand: A c∧ B,
uiff: uiff(P;Q)
Lemmas referenced :
geo-lt-angle_wf,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-congruent-preserves-strict-between,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-between-inner-trans,
geo-between-exchange3,
geo-between-exchange4,
geo-between-outer-trans,
geo-sep-sym,
geo-between-sep,
geo-strict-between-sep2,
geo-strict-between-sep3,
geo-congruent-iff-length,
geo-length-flip
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
universeIsType,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
productElimination,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (a-b-c {}\mRightarrow{} x-y-z {}\mRightarrow{} (\mneg{}abc < xyz))
Date html generated:
2019_10_16-PM-01_49_39
Last ObjectModification:
2019_09_27-PM-06_01_41
Theory : euclidean!plane!geometry
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