Nuprl Lemma : geo-perp-in-same-colinear
∀[e:BasicGeometry]. ∀[a,b,c,d,x:Point]. (Colinear(x;c;d)) supposing (ab ⊥x xc and ab ⊥x xd)
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
geo-colinear: Colinear(a;b;c),
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
basic-geometry: BasicGeometry,
stable: Stable{P},
or: P ∨ Q,
geo-eq: a ≡ b,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
cand: A c∧ B,
geo-perp-in: ab ⊥x cd,
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,
right-angle: Rabc,
geo-midpoint: a=m=b
Lemmas referenced :
not_wf,
geo-between_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-perp-in_wf,
geo-point_wf,
stable__not,
false_wf,
or_wf,
geo-sep_wf,
minimal-double-negation-hyp-elim,
geo-perp-in_functionality,
geo-eq_weakening,
minimal-not-not-excluded-middle,
geo-sep-or,
symmetric-point-construction,
geo-sep-sym,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf,
right-angle-symmetry,
geo-midpoint-symmetry,
upper-dimension-axiom,
geo-congruent-iff-length,
geo-length-flip,
geo-between-sep,
geo-perp-in-not-eq,
geo-eq-self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
productEquality,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
functionEquality,
independent_functionElimination,
lambdaFormation,
unionElimination,
productElimination,
setElimination,
rename,
dependent_set_memberEquality,
voidEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[a,b,c,d,x:Point]. (Colinear(x;c;d)) supposing (ab \mbot{}x xc and ab \mbot{}x xd)
Date html generated:
2018_05_22-PM-00_05_26
Last ObjectModification:
2018_05_11-PM-06_45_17
Theory : euclidean!plane!geometry
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