Nuprl Lemma : between-preserves-left-3
∀e:EuclideanPlane. ∀A,B,C,V:Point. (C leftof AB ⇒ A ≠ V ⇒ V_A_B ⇒ C leftof VA)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-left: a leftof bc,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
oriented-plane: OrientedPlane,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-sep-sym,
geo-between-symmetry,
left-all-symmetry,
geo-point_wf,
all_wf,
geo-sep_wf,
geo-between_wf,
geo-left_wf,
left-between-implies-right1
Rules used in proof :
instantiate,
independent_isectElimination,
independent_pairFormation,
functionEquality,
lambdaEquality,
because_Cache,
applyEquality,
isectElimination,
productEquality,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
sqequalRule,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A,B,C,V:Point. (C leftof AB {}\mRightarrow{} A \mneq{} V {}\mRightarrow{} V\_A\_B {}\mRightarrow{} C leftof VA)
Date html generated:
2019_10_16-PM-01_32_32
Last ObjectModification:
2018_10_24-PM-02_05_35
Theory : euclidean!plane!geometry
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