Nuprl Lemma : geo-eq-implies-col
∀e:BasicGeometry. ∀a,b,c:Point. ((¬(a = b ∈ Point)) ⇒ (b = c ∈ Point) ⇒ Colinear(a;b;c))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
true: True,
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
squash: ↓T,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
not_wf,
Error :basic-geo-primitives_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
equal_wf,
geo-colinear-same,
iff_weakening_equal,
Error :basic-geo-structure_wf,
geo-point_wf,
true_wf,
squash_wf,
geo-colinear_wf
Rules used in proof :
instantiate,
because_Cache,
independent_functionElimination,
productElimination,
independent_isectElimination,
universeEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
rename,
setElimination,
sqequalRule,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. ((\mneg{}(a = b)) {}\mRightarrow{} (b = c) {}\mRightarrow{} Colinear(a;b;c))
Date html generated:
2017_10_02-PM-06_39_59
Last ObjectModification:
2017_08_05-PM-04_46_40
Theory : euclidean!plane!geometry
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