Nuprl Lemma : eu-perp-three
∀e:EuclideanPlane. ∀x,a,b,c:Point. (Perp-in(x; ba; ca) ⇒ (x = a ∈ Point))
Proof
Definitions occuring in Statement :
eu-perp-in: Perp-in(x; ab; cd),
euclidean-plane: EuclideanPlane,
eu-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
eu-perp-in: Perp-in(x; ab; cd),
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
iff: P ⇐⇒ Q,
eu-colinear-set: eu-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),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
eu-perpendicular: Per(a;b;c),
exists: ∃x:A. B[x],
uimplies: b supposing a,
eu-midpoint: a=m=b
Lemmas referenced :
eu-between-eq-same2,
eu-congruence-identity-sym,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
eu-colinear-is-colinear-set,
eu-colinear-def,
euclidean-plane_wf,
eu-point_wf,
eu-perp-in_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
setElimination,
rename,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
introduction,
imageMemberEquality,
baseClosed,
independent_isectElimination,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}x,a,b,c:Point. (Perp-in(x; ba; ca) {}\mRightarrow{} (x = a))
Date html generated:
2016_05_18-AM-06_43_34
Last ObjectModification:
2016_01_16-PM-10_28_45
Theory : euclidean!geometry
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