Nuprl Lemma : colinear-right-angle
∀e:BasicGeometry. ∀a,b,c:Point. (b ≠ c ⇒ Colinear(a;b;c) ⇒ Rabc ⇒ a ≡ b)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
right-angle: Rabc,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-eq: a ≡ b,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
and: P ∧ Q,
cand: A c∧ B,
basic-geometry: BasicGeometry,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
right-angle_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear_wf,
geo-sep_wf,
geo-point_wf,
stable__not,
false_wf,
or_wf,
not_wf,
geo-eq_inversion,
right-angle-legs-same,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-sep-sym,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf,
right-angle-colinear2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
voidElimination,
independent_pairFormation,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (b \mneq{} c {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} Rabc {}\mRightarrow{} a \mequiv{} b)
Date html generated:
2018_05_22-PM-00_02_57
Last ObjectModification:
2018_04_02-AM-11_15_27
Theory : euclidean!plane!geometry
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