Nuprl Lemma : Euclid-Prop1
∀e:EuclideanPlane. ∀a,b:Point. (a ≠ b ⇒ (∃c:Point. EQΔ(c;b;a)))
Proof
Definitions occuring in Statement :
geo-equilateral: EQΔ(a;b;c),
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
squash: ↓T,
guard: {T},
sq_stable: SqStable(P),
cand: A c∧ B,
and: P ∧ Q,
geo-equilateral: EQΔ(a;b;c),
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
sq_stable__geo-congruent,
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-equilateral_wf,
lsep-all-sym2,
sq_stable__geo-lsep,
geo-sep_wf,
Euclid-Prop1-left-ext
Rules used in proof :
independent_isectElimination,
instantiate,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
dependent_pairFormation,
rename,
setElimination,
sqequalRule,
applyEquality,
isectElimination,
hypothesis,
because_Cache,
dependent_set_memberEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. (a \mneq{} b {}\mRightarrow{} (\mexists{}c:Point. EQ\mDelta{}(c;b;a)))
Date html generated:
2018_05_22-AM-11_55_08
Last ObjectModification:
2018_05_21-AM-01_14_01
Theory : euclidean!plane!geometry
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