Nuprl Lemma : colinear-right-angle-trivial
∀e:BasicGeometry. ∀a,b,c:Point. (Rabc ⇒ Colinear(a;b;c) ⇒ (¬(a ≠ b ∧ c ≠ b)))
Proof
Definitions occuring in Statement :
right-angle: Rabc,
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: 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,
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
iff: P ⇐⇒ Q
Lemmas referenced :
geo-point_wf,
right-angle_wf,
geo-colinear_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
right-angle-colinear,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geo-sep-irrefl',
geo-eq_weakening,
geo-sep_functionality,
right-angle-legs-same
Rules used in proof :
rename,
setElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
productEquality,
voidElimination,
independent_functionElimination,
hypothesis,
because_Cache,
productElimination,
sqequalHypSubstitution,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
dependent_functionElimination,
promote_hyp
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (Rabc {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} (\mneg{}(a \mneq{} b \mwedge{} c \mneq{} b)))
Date html generated:
2017_10_02-PM-06_42_10
Last ObjectModification:
2017_08_05-PM-04_48_23
Theory : euclidean!plane!geometry
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