Nuprl Lemma : not-dist-lemma
∀g:EuclideanPlane. ∀a,b,c,d:Point. ((¬D(a;b;b;b;c;d)) ⇒ (¬ab > cd))
Proof
Definitions occuring in Statement :
dist: D(a;b;c;d;e;f),
euclidean-plane: EuclideanPlane,
geo-gt: cd > ab,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
geo-gt: cd > ab,
dist: D(a;b;c;d;e;f),
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
cand: A c∧ B,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
geo-gt_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
not_wf,
dist_wf,
geo-point_wf,
geo-sep-sym,
geo-sep_wf,
geo-between-trivial,
geo-congruent-refl,
geo-congruent-flip,
geo-congruent-iff-length,
geo-between_wf,
geo-congruent_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalHypSubstitution,
imageElimination,
independent_functionElimination,
productElimination,
voidElimination,
hypothesis,
because_Cache,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_pairFormation,
dependent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
equalitySymmetry,
productEquality,
setElimination,
rename,
setEquality,
lambdaEquality
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. ((\mneg{}D(a;b;b;b;c;d)) {}\mRightarrow{} (\mneg{}ab > cd))
Date html generated:
2019_10_16-PM-02_48_09
Last ObjectModification:
2018_09_15-PM-03_16_12
Theory : euclidean!plane!geometry
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