Nuprl Lemma : geo-gt-out-to-between
∀e:EuclideanPlane. ∀a,b,c:Point. (out(a bc) ⇒ ac > ab ⇒ a-b-c)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-gt: cd > ab,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-strict-between: a-b-c,
and: P ∧ Q,
cand: A c∧ B,
geo-out: out(p ab),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
stable: Stable{P},
not: ¬A,
basic-geometry-: BasicGeometry-,
false: False,
geo-gt: cd > ab,
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q)
Lemmas referenced :
geo-gt_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-out_wf,
geo-point_wf,
stable__geo-between,
double-negation-hyp-elim,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
not_wf,
geo-between_wf,
geo-lt-out-to-between,
geo-strict-between-implies-between,
istype-void,
geo-gt-implies-lt,
sq_stable__geo-sep,
geo-sep_functionality,
geo-eq_weakening,
stable_geo-eq,
geo-eq_wf,
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
geo-sep-sym,
geo-sep-O-X,
geo-construction-unicity-from-first,
euclidean-plane-axioms,
geo-strict-between-sep3,
geo-between-symmetry,
geo-between-exchange3,
geo-between-exchange4,
geo-between-inner-trans,
geo-between-outer-trans,
geo-congruent-sep,
geo-add-length-between,
geo-congruent-iff-length,
geo-add-length_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
because_Cache,
dependent_functionElimination,
setElimination,
rename,
independent_functionElimination,
voidElimination,
functionIsType,
imageElimination,
imageMemberEquality,
baseClosed,
equalitySymmetry,
dependent_set_memberEquality_alt,
equalityTransitivity,
productIsType,
equalityIstype,
applyLambdaEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (out(a bc) {}\mRightarrow{} ac > ab {}\mRightarrow{} a-b-c)
Date html generated:
2019_10_16-PM-01_24_19
Last ObjectModification:
2019_06_18-AM-11_05_40
Theory : euclidean!plane!geometry
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