Nuprl Lemma : geo-out-interior-point-exists
∀g:EuclideanPlane. ∀a,b,c,a',c',x:Point.
(a # bc
⇒ out(b aa')
⇒ out(b cc')
⇒ a-x-c
⇒ (∃x':Point. (((a'-x'-c' ∧ out(b xx')) ∧ abx ≅a a'bx') ∧ cbx ≅a c'bx')))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
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,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-lsep: a # bc,
or: P ∨ Q,
prop: ℙ,
oriented-plane: OrientedPlane,
basic-geometry-: BasicGeometry-,
exists: ∃x:A. B[x]
Lemmas referenced :
colinear-lsep,
lsep-all-sym,
geo-sep-sym,
geo-strict-between-sep2,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
geo-strict-between_wf,
geo-out_wf,
geo-lsep_wf,
geo-point_wf,
left-between-implies-right1,
geo-between-symmetry,
geo-strict-between-implies-between,
euclidean-plane-axioms,
geo-strict-between-sep3,
geo-left-out-2,
geo-left-out-3,
use-plane-sep_strict,
geo-cong-angle_wf,
left-all-symmetry,
left-convex2,
geo-between_wf,
geo-sep_wf,
geo-colinear-left-out3,
geo-strict-between-sym,
geo-left-out-4,
geo-cong-angle-refl,
geo-out_weakening,
left-implies-sep,
geo-eq_weakening,
lsep-implies-sep,
out-preserves-angle-cong_1,
left-between-implies-right2,
left-convex
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,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
unionElimination,
universeIsType,
inhabitedIsType,
rename,
dependent_pairFormation_alt,
inlFormation_alt
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,a',c',x:Point.
(a \# bc
{}\mRightarrow{} out(b aa')
{}\mRightarrow{} out(b cc')
{}\mRightarrow{} a-x-c
{}\mRightarrow{} (\mexists{}x':Point. (((a'-x'-c' \mwedge{} out(b xx')) \mwedge{} abx \mcong{}\msuba{} a'bx') \mwedge{} cbx \mcong{}\msuba{} c'bx')))
Date html generated:
2019_10_16-PM-02_00_24
Last ObjectModification:
2018_10_25-AM-10_03_49
Theory : euclidean!plane!geometry
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