Nuprl Lemma : euclid-Prop3
∀e:EuclideanPlane. ∀A:Point. ∀B:{B:Point| A ≠ B} . ∀C1:Point. ∀C2:{C2:Point| C1 ≠ C2} .
(|C1C2| < |AB| ⇒ (∃E:{Point| (A_E_B ∧ AE ≅ C1C2)}))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
basic-geometry-: BasicGeometry-,
sq_exists: ∃x:{A| B[x]},
false: False,
not: ¬A,
stable: Stable{P},
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True
Lemmas referenced :
sq_stable__geo-sep,
sq_stable__and,
geo-sep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
set_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-lt_wf,
geo-sep-sym,
geo-extend-exists,
geo-congruent-sep,
geo-congruent_wf,
geo-between_wf,
geo-between-symmetry,
geo-between-same-side2,
not_wf,
stable__geo-between,
geo-add-length-between,
geo-add-length_wf,
geo-length-type_wf,
equal_wf,
geo-congruent-iff-length,
iff_weakening_equal,
basic-geometry_wf,
true_wf,
squash_wf,
geo-lt_transitivity,
geo-le-add1,
geo-lt-irrefl
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
isect_memberEquality,
lambdaEquality,
independent_isectElimination,
instantiate,
applyEquality,
because_Cache,
hypothesis,
hypothesisEquality,
sqequalRule,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
rename,
thin,
setElimination,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
productEquality,
dependent_set_memberFormation,
voidElimination,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
universeEquality,
natural_numberEquality,
equalityTransitivity
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A:Point. \mforall{}B:\{B:Point| A \mneq{} B\} . \mforall{}C1:Point. \mforall{}C2:\{C2:Point| C1 \mneq{} C2\} .
(|C1C2| < |AB| {}\mRightarrow{} (\mexists{}E:\{Point| (A\_E\_B \mwedge{} AE \00D0 C1C2)\}))
Date html generated:
2017_10_02-PM-06_56_07
Last ObjectModification:
2017_08_06-PM-08_41_36
Theory : euclidean!plane!geometry
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