Nuprl Lemma : sq_stable__geo-left-axioms
∀g:EuclideanPlane
SqStable((∀a,b,c:Point. (¬a # bc ⇐⇒ Colinear(a;b;c)))
∧ (∀a,b,c:Point. (a leftof bc ⇒ b leftof ca))
∧ (∀a,b,c:Point. (a leftof bc ⇒ b # c))
∧ (∀a,b,x,y,z:Point. (x leftof ab ⇒ y leftof ab ⇒ B(xzy) ⇒ z # ab))
∧ (∀a,b,c,y:Point. (a # bc ⇒ y # b ⇒ Colinear(y;a;b) ⇒ y # bc)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-between: B(abc),
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-sep: a # b,
geo-point: Point,
sq_stable: SqStable(P),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
euclidean-plane: EuclideanPlane,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
sq_stable__geo-lsep,
sq_stable__geo-sep,
sq_stable__geo-left,
sq_stable__colinear,
sq_stable__not,
sq_stable__iff,
sq_stable__all,
istype-void,
geo-between_wf,
geo-sep_wf,
geo-left_wf,
geo-colinear_wf,
geo-lsep_wf,
not_wf,
iff_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
sq_stable__and
Rules used in proof :
rename,
setElimination,
dependent_functionElimination,
lambdaEquality_alt,
independent_functionElimination,
productIsType,
inhabitedIsType,
universeIsType,
functionIsType,
productEquality,
isect_memberEquality_alt,
because_Cache,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
functionEquality,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane
SqStable((\mforall{}a,b,c:Point. (\mneg{}a \# bc \mLeftarrow{}{}\mRightarrow{} Colinear(a;b;c)))
\mwedge{} (\mforall{}a,b,c:Point. (a leftof bc {}\mRightarrow{} b leftof ca))
\mwedge{} (\mforall{}a,b,c:Point. (a leftof bc {}\mRightarrow{} b \# c))
\mwedge{} (\mforall{}a,b,x,y,z:Point. (x leftof ab {}\mRightarrow{} y leftof ab {}\mRightarrow{} B(xzy) {}\mRightarrow{} z \# ab))
\mwedge{} (\mforall{}a,b,c,y:Point. (a \# bc {}\mRightarrow{} y \# b {}\mRightarrow{} Colinear(y;a;b) {}\mRightarrow{} y \# bc)))
Date html generated:
2019_10_30-AM-06_18_27
Last ObjectModification:
2019_10_29-PM-02_52_38
Theory : euclidean!plane!geometry
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