Nuprl Lemma : geo-out-le-iff-bet
∀e:BasicGeometry. ∀a,b,c:Point. (out(a bc) ⇒ (|ab| ≤ |ac| ⇐⇒ a_b_c))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-le: p ≤ q,
geo-length: |s|,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
false: False,
cand: A c∧ B,
not: ¬A,
stable: Stable{P},
geo-out: out(p ab),
true: True,
squash: ↓T
Lemmas referenced :
geo-point_wf,
geo-out_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-between_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-le_wf,
not_wf,
stable__geo-between,
geo-add-length-between,
iff_weakening_equal,
geo-length-type_wf,
true_wf,
squash_wf,
geo-eq_weakening,
geo-between_functionality,
geo-add-length-le-implies-eq,
geo-le-add1
Rules used in proof :
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
because_Cache,
hypothesis,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidElimination,
independent_functionElimination,
dependent_functionElimination,
productElimination,
universeEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
promote_hyp
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (out(a bc) {}\mRightarrow{} (|ab| \mleq{} |ac| \mLeftarrow{}{}\mRightarrow{} a\_b\_c))
Date html generated:
2017_10_02-PM-06_27_41
Last ObjectModification:
2017_08_05-PM-04_40_48
Theory : euclidean!plane!geometry
Home
Index