Nuprl Lemma : right-angle-colinear
∀e:BasicGeometry. ∀a,b,c,a':Point. (Rabc ⇒ a ≠ b ⇒ Colinear(b;a;a') ⇒ Ra'bc)
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],
implies: 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,
right-angle: Rabc,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
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),
and: P ∧ Q,
geo-midpoint: a=m=b
Lemmas referenced :
geo-point_wf,
right-angle_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-colinear_wf,
geo-midpoint_wf,
geo-length-flip,
geo-congruent-iff-length,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geo-colinear-congruence1
Rules used in proof :
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
rename,
setElimination,
because_Cache,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
levelHypothesis,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_functionElimination,
dependent_functionElimination,
productElimination,
addLevel
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a':Point. (Rabc {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} Colinear(b;a;a') {}\mRightarrow{} Ra'bc)
Date html generated:
2017_10_02-PM-06_41_18
Last ObjectModification:
2017_08_05-PM-04_47_47
Theory : euclidean!plane!geometry
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