Nuprl Lemma : between-preserves-left-5
∀e:EuclideanPlane. ∀A,B,C,V:Point. (C leftof AB ⇒ B ≠ V ⇒ A_V_B ⇒ C leftof VB)
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
geo-out: out(p ab)
Lemmas referenced :
euclidean-plane-axioms,
geo-sep-sym,
left-implies-sep,
geo-left_wf,
geo-sep_wf,
istype-void,
geo-eq_wf,
geo-between_wf,
geo-congruent_wf,
geo-ge_wf,
geo-lsep_wf,
geo-colinear_wf,
geo-between-out,
geo-out_wf,
geo-between-symmetry,
geo-left-out-better-1,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
hypothesis,
independent_functionElimination,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
sqequalRule,
productIsType,
functionIsType,
independent_pairFormation,
independent_isectElimination,
instantiate,
inhabitedIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A,B,C,V:Point. (C leftof AB {}\mRightarrow{} B \mneq{} V {}\mRightarrow{} A\_V\_B {}\mRightarrow{} C leftof VB)
Date html generated:
2019_10_16-PM-01_32_53
Last ObjectModification:
2018_11_27-PM-00_01_58
Theory : euclidean!plane!geometry
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