Nuprl Lemma : geo-lt-pt-exists
∀e:BasicGeometry. ∀a,b,c,d:Point. (a ≠ b ⇒ ab < cd ⇒ (∃x:Point. (a-b-x ∧ ax ≅ cd)))
Proof
Definitions occuring in Statement :
geo-lt-pt: ab < cd,
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-lt-pt: ab < cd,
implies: P ⇒ Q,
all: ∀x:A. B[x],
true: True,
squash: ↓T,
uiff: uiff(P;Q),
cand: A c∧ B
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-lt-pt_wf,
geo-strict-between-sep3,
geo-proper-extend-exists,
geo-add-length-between,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-mk-seg_wf,
geo-length_wf,
geo-add-length_wf,
geo-length-type_wf,
equal_wf,
and_wf,
geo-congruent-iff-length,
true_wf,
squash_wf,
geo-congruent_wf,
geo-strict-between_wf
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
lambdaEquality,
hyp_replacement,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
dependent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (a \mneq{} b {}\mRightarrow{} ab < cd {}\mRightarrow{} (\mexists{}x:Point. (a-b-x \mwedge{} ax \00D0 cd)))
Date html generated:
2017_10_02-PM-06_36_20
Last ObjectModification:
2017_08_05-PM-04_45_22
Theory : euclidean!plane!geometry
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