Nuprl Lemma : Euclid-drop-perp-1
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . ∀c:{c:Point| ∀x:Point. (Colinear(a;b;x) ⇒ c ≠ x)} .
∃p:Point. (Colinear(a;b;p) ∧ ab ⊥p pc)
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
sq_stable: SqStable(P),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
geo-perp-in: ab ⊥x cd,
squash: ↓T,
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),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
Euclid-drop-perp-0,
sq_stable__colinear,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
euclidean-plane-axioms,
sq_stable__from_stable,
geo-perp-in_wf,
stable__geo-perp-in,
geo-perp-in-iff2,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-colinear_wf,
geo-point_wf,
all_wf,
geo-sep_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
productElimination,
dependent_pairFormation,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
productEquality,
setEquality,
lambdaEquality,
functionEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} .
\mforall{}c:\{c:Point| \mforall{}x:Point. (Colinear(a;b;x) {}\mRightarrow{} c \mneq{} x)\} .
\mexists{}p:Point. (Colinear(a;b;p) \mwedge{} ab \mbot{}p pc)
Date html generated:
2018_05_22-PM-00_11_58
Last ObjectModification:
2018_05_11-PM-03_22_52
Theory : euclidean!plane!geometry
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