Nuprl Lemma : center-on-circle-overlap
∀e:EuclideanPlane. ∀a,b,c:Point. (a ≠ b ⇒ |bc| ≤ |ab| + |ab| ⇒ Overlap(a;b;b;c))
Proof
Definitions occuring in Statement :
geo-add-length: p + q,
geo-le: p ≤ q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
circle-overlap: Overlap(a;b;c;d),
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,
circle-overlap: Overlap(a;b;c;d),
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
geo-ge: cd ≥ ab,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
geo-eq: a ≡ b,
basic-geometry-: BasicGeometry-,
true: True
Lemmas referenced :
geo-SCS_wf,
geo-sep-sym,
geo-between-trivial2,
geo-sep_wf,
geo-between_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-congruent_wf,
geo-SCO_wf,
geo-colinear_wf,
geo-congruent-refl,
geo-ge-trivial2,
geo-congruent-iff-length,
geo-SCS-congruent,
geo-ge_wf,
exists_wf,
geo-le_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
stable__not,
not_wf,
false_wf,
or_wf,
geo-sep_functionality,
geo-eq_weakening,
minimal-double-negation-hyp-elim,
geo-congruent_functionality,
geo-ge_functionality,
minimal-not-not-excluded-middle,
geo-SCS-out,
equal_wf,
geo-out_wf,
set_wf,
geo-out-iff-colinear,
geo-between-symmetry,
geo-between-implies-ge,
stable__geo-ge,
geo-simple-colinear-cases,
geo-add-length-between,
iff_weakening_equal,
geo-length-type_wf,
true_wf,
squash_wf,
geo-length-flip,
basic-geometry_wf,
geo-add-length-comm,
geo-add-length-cancel-left-le,
geo-le-iff,
geo-ge-trivial,
geo-congruence-identity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
dependent_pairFormation,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
setEquality,
instantiate,
independent_isectElimination,
functionEquality,
productElimination,
imageMemberEquality,
baseClosed,
unionElimination,
voidElimination,
natural_numberEquality,
universeEquality,
imageElimination,
promote_hyp
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (a \mneq{} b {}\mRightarrow{} |bc| \mleq{} |ab| + |ab| {}\mRightarrow{} Overlap(a;b;b;c))
Date html generated:
2018_05_22-PM-00_00_35
Last ObjectModification:
2018_03_30-AM-10_52_27
Theory : euclidean!plane!geometry
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