Nuprl Lemma : sympoint_wf
∀[e:EuclideanPlane]. ∀[a:Point]. ∀[p:{p:Point| a ≠ p} ]. (SymmetricPoint(a;p) ∈ {p':Point| p=a=p'} )
Proof
Definitions occuring in Statement :
sympoint: SymmetricPoint(a;p),
euclidean-plane: EuclideanPlane,
geo-midpoint: a=m=b,
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sympoint: SymmetricPoint(a;p),
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
geo-midpoint: a=m=b
Lemmas referenced :
geo-SCO_wf,
sq_stable__geo-sep,
geo-between-trivial2,
geo-sep_wf,
geo-between_wf,
set_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-midpoint_wf,
geo-congruent_wf,
geo-congruent-flip,
geo-congruent-symmetry,
geo-congruent-transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
productElimination,
setEquality,
functionEquality
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[a:Point]. \mforall{}[p:\{p:Point| a \mneq{} p\} ]. (SymmetricPoint(a;p) \mmember{} \{p':Point| p=a=p'\} \000C)
Date html generated:
2018_05_22-AM-11_53_35
Last ObjectModification:
2018_03_30-PM-04_52_27
Theory : euclidean!plane!geometry
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