Nuprl Lemma : cong-angle-preserves-lsep_weak
∀g:EuclideanPlane. ∀a,b,c,x,y,z:Point. (x # yz ⇒ abc ≅a xyz ⇒ (¬¬a # bc))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
basic-geometry-: BasicGeometry-,
geo-cong-angle: abc ≅a xyz,
geo-eq: a ≡ b,
oriented-plane: OrientedPlane,
geo-strict-between: a-b-c
Lemmas referenced :
not-lsep-iff-colinear,
geo-colinear-symmetry,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
istype-void,
geo-cong-angle_wf,
geo-point_wf,
geo-colinear-cases,
false_wf,
stable__false,
geo-eq_wf,
geo-strict-between_wf,
geo-out_weakening,
geo-eq_weakening,
geo-cong-angle-symm2,
not-lsep-if-out,
geo-cong-angle_functionality,
angle-cong-preserves-zero-angle,
angle-cong-preserves-straight-angle,
geo-between-symmetry,
geo-strict-between-implies-between,
lsep-not-between,
geo-between-out,
euclidean-plane-axioms,
geo-sep-sym,
geo-strict-between-sep2,
geo-strict-between-sep1,
geo-out_inversion,
geo-strict-between-sep3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
isectElimination,
sqequalRule,
independent_isectElimination,
because_Cache,
independent_pairFormation,
voidElimination,
functionIsType,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (x \# yz {}\mRightarrow{} abc \mcong{}\msuba{} xyz {}\mRightarrow{} (\mneg{}\mneg{}a \# bc))
Date html generated:
2019_10_16-PM-01_58_00
Last ObjectModification:
2018_11_08-PM-02_04_37
Theory : euclidean!plane!geometry
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