Nuprl Lemma : geo-gt-not-congruent
∀g:EuclideanPlane. ∀a,b,c,d:Point. (ab > cd ⇒ (¬ab ≅ cd))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-gt: cd > ab,
geo-congruent: ab ≅ cd,
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,
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
geo-eq: a ≡ b,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ
Lemmas referenced :
geo-between-congruent,
geo-congruent-iff-length,
geo-sep-sym,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-gt_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
sqequalHypSubstitution,
imageElimination,
productElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
isectElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (ab > cd {}\mRightarrow{} (\mneg{}ab \mcong{} cd))
Date html generated:
2019_10_16-PM-01_39_35
Last ObjectModification:
2019_07_16-PM-01_31_22
Theory : euclidean!plane!geometry
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