Nuprl Lemma : geo-triangle-implies
∀e:HeytingGeometry. ∀a,b,c:Point.
(a # bc ⇒ {c # ba ∧ c # ab ∧ a ≠ c ∧ (¬a_b_c) ∧ (∀z:Point. (z ≠ b ⇒ Colinear(a;b;z) ⇒ z # bc))})
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
geo-triangle: a # bc,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
member: t ∈ T,
heyting-geometry: HeytingGeometry,
cand: A c∧ 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,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
uall: ∀[x:A]. B[x],
select: L[n],
cons: [a / b],
subtract: n - m,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
oriented-plane: OrientedPlane
Lemmas referenced :
lsep-all-sym,
lsep-implies-sep,
colinear-lsep-cycle,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-colinear_wf,
geo-sep_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
lsep-not-between,
subtype_rel_self,
basic-geo-axioms_wf,
geo-left-axioms_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
setEquality,
productEquality,
cumulativity
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point.
(a \# bc
{}\mRightarrow{} \{c \# ba \mwedge{} c \# ab \mwedge{} a \mneq{} c \mwedge{} (\mneg{}a\_b\_c) \mwedge{} (\mforall{}z:Point. (z \mneq{} b {}\mRightarrow{} Colinear(a;b;z) {}\mRightarrow{} z \# bc))\})
Date html generated:
2017_10_02-PM-07_01_22
Last ObjectModification:
2017_08_09-PM-05_30_17
Theory : euclidean!plane!geometry
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