Nuprl Lemma : geo-out-strict_weakening
∀e:BasicGeometry. ∀a,b,c:Point. (Colinear(a;b;c) ⇒ (¬c_a_b) ⇒ a ≠ b ⇒ b ≠ c ⇒ geo-out-strict(e;a;b;c))
Proof
Definitions occuring in Statement :
geo-out-strict: geo-out-strict(e;p;a;b),
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
basic-geometry: BasicGeometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
not: ¬A,
geo-out-strict: geo-out-strict(e;p;a;b),
implies: P ⇒ Q,
all: ∀x:A. B[x],
false: False,
cand: A c∧ B,
geo-strict-between: a-b-c,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
or: P ∨ Q,
stable: Stable{P}
Lemmas referenced :
geo-point_wf,
geo-colinear_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-strict-between_wf,
not_wf,
geo-colinear-implies,
minimal-not-not-excluded-middle,
geo-between-trivial2,
geo-between_functionality,
geo-sep_functionality,
geo-eq_weakening,
geo-strict-between_functionality,
minimal-double-negation-hyp-elim,
or_wf,
false_wf,
stable__not,
geo-sep-sym,
geo-between-symmetry
Rules used in proof :
rename,
setElimination,
independent_isectElimination,
instantiate,
sqequalRule,
hypothesis,
because_Cache,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
productEquality,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination,
voidElimination,
independent_pairFormation,
impliesLevelFunctionality,
promote_hyp,
levelHypothesis,
impliesFunctionality,
addLevel,
unionElimination,
functionEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point.
(Colinear(a;b;c) {}\mRightarrow{} (\mneg{}c\_a\_b) {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} geo-out-strict(e;a;b;c))
Date html generated:
2017_10_02-PM-06_25_42
Last ObjectModification:
2017_08_05-PM-04_19_14
Theory : euclidean!plane!geometry
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