Nuprl Lemma : geo-out-preserves-lt-angle
∀e:EuclideanPlane. ∀a,b,c,a',c',x,y,z:Point. (a # bc ⇒ out(b aa') ⇒ out(b cc') ⇒ abc < xyz ⇒ a'bc' < xyz)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q
Lemmas referenced :
geo-lt-angle_wf,
geo-out_wf,
geo-lsep_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-cong-angle-refl,
lsep-implies-sep,
geo-out_weakening,
geo-eq_weakening,
geo-sep-sym,
geo-cong-angle-symm2,
out-preserves-angle-cong_1,
geo-cong-angle-preserves-lt-angle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
sqequalRule,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
because_Cache,
independent_functionElimination,
productElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,a',c',x,y,z:Point.
(a \# bc {}\mRightarrow{} out(b aa') {}\mRightarrow{} out(b cc') {}\mRightarrow{} abc < xyz {}\mRightarrow{} a'bc' < xyz)
Date html generated:
2019_10_16-PM-02_00_09
Last ObjectModification:
2019_01_04-PM-01_37_02
Theory : euclidean!plane!geometry
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