Nuprl Lemma : eqtri_wf
∀[e:EuclideanPlane]. ∀[a:Point]. ∀[b:{b:Point| a ≠ b} ].
(Δ(a;b) ∈ {c:Point| ((cb ≅ ab ∧ ca ≅ ba) ∧ ca ≅ cb) ∧ c leftof ab} )
Proof
Definitions occuring in Statement :
eqtri: Δ(a;b),
euclidean-plane: EuclideanPlane,
geo-left: a leftof bc,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
sq_exists: ∃x:A [B[x]],
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
geo-CC-2,
Euclid-Prop1-left,
record-select: r.x,
geo-CC: CC(a;b;c;d),
geo-CCL: CCL(a;b;c;d),
eqtri: Δ(a;b),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
geo-primitives_wf,
euclidean-plane-structure_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
set_wf,
geo-left_wf,
geo-congruent_wf,
sq_exists_wf,
geo-sep_wf,
geo-point_wf,
all_wf,
euclidean-plane_wf,
subtype_rel_self,
Euclid-Prop1-left,
geo-CC-2
Rules used in proof :
isect_memberEquality,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
rename,
setElimination,
productEquality,
setEquality,
lambdaEquality,
because_Cache,
hypothesisEquality,
cumulativity,
functionEquality,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
instantiate,
thin,
applyEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[a:Point]. \mforall{}[b:\{b:Point| a \mneq{} b\} ].
(\mDelta{}(a;b) \mmember{} \{c:Point| ((cb \mcong{} ab \mwedge{} ca \mcong{} ba) \mwedge{} ca \mcong{} cb) \mwedge{} c leftof ab\} )
Date html generated:
2018_05_22-AM-11_53_42
Last ObjectModification:
2018_05_21-AM-01_13_22
Theory : euclidean!plane!geometry
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