Nuprl Lemma : geo-triangle-colinear3
∀e:HeytingGeometry. ∀a,b,c,x,y,z:Point.
(a # bc ⇒ x ≠ b ⇒ Colinear(a;b;x) ⇒ y ≠ c ⇒ Colinear(b;c;y) ⇒ z ≠ x ⇒ Colinear(c;x;z) ⇒ x # yz)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
heyting-geometry: Error :heyting-geometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q
Lemmas referenced :
geo-point_wf,
Error :geo-triangle_wf,
Error :basic-geo-primitives_wf,
geo-sep_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
Error :heyting-geometry_wf,
subtype_rel_transitivity,
heyting-geometry-subtype,
basic-geometry-subtype,
geo-colinear_wf,
geo-triangle-colinear2,
geo-triangle-symmetry,
geo-triangle-colinear
Rules used in proof :
rename,
setElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,x,y,z:Point.
(a \# bc
{}\mRightarrow{} x \mneq{} b
{}\mRightarrow{} Colinear(a;b;x)
{}\mRightarrow{} y \mneq{} c
{}\mRightarrow{} Colinear(b;c;y)
{}\mRightarrow{} z \mneq{} x
{}\mRightarrow{} Colinear(c;x;z)
{}\mRightarrow{} x \# yz)
Date html generated:
2017_10_02-PM-07_02_03
Last ObjectModification:
2017_08_06-PM-08_55_11
Theory : euclidean!plane!geometry
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