Nuprl Lemma : Euclid-Prop16-construction-lemma
∀g:EuclideanPlane. ∀a,b,c:Point.
(a # bc ⇒ (∃x:{x:Point| b=x=c} . ∃y:{y:Point| a=x=y} . (abc ≅a ycb ∧ (b ≠ x ∧ x ≠ c) ∧ a ≠ x ∧ x ≠ y)))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-midpoint: a=m=b,
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],
implies: P ⇒ Q,
member: t ∈ T,
guard: {T},
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
cand: A c∧ B,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
geo-midpoint: a=m=b,
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,
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
geo-strict-between: a-b-c,
geo-cong-angle: abc ≅a xyz
Lemmas referenced :
Euclid-midpoint,
lsep-implies-sep,
geo-sep_wf,
sq_stable__midpoint,
geo-midpoint_wf,
colinear-lsep,
lsep-all-sym,
geo-sep-sym,
midpoint-sep,
geo-colinear-permute,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
geo-cong-angle_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
colinear-implies-midpoint,
geo-strict-between-sep1,
geo-strict-between-implies-colinear,
geo-congruent-iff-length,
geo-length-flip,
geo-strict-between-sep3,
geo-proper-extend-exists,
geo-strict-between-implies-between,
geo-between-symmetry,
euclidean-plane-axioms,
geo-strict-between-sep2,
geo-between-trivial,
vertical-angles-congruent,
geo-between_wf,
geo-congruent_wf,
geo-sas2,
geo-between-out,
geo-out_weakening,
geo-eq_weakening,
out-preserves-angle-cong_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
independent_functionElimination,
hypothesis,
productElimination,
dependent_set_memberEquality_alt,
universeIsType,
isectElimination,
applyEquality,
sqequalRule,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation_alt,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
independent_pairFormation,
productIsType,
setIsType,
instantiate,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c:Point.
(a \# bc
{}\mRightarrow{} (\mexists{}x:\{x:Point| b=x=c\} . \mexists{}y:\{y:Point| a=x=y\} . (abc \mcong{}\msuba{} ycb \mwedge{} (b \mneq{} x \mwedge{} x \mneq{} c) \mwedge{} a \mneq{} x \mwedge{} x \mneq{} y)))
Date html generated:
2019_10_16-PM-02_14_38
Last ObjectModification:
2018_12_03-PM-09_58_16
Theory : euclidean!plane!geometry
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