Nuprl Lemma : geo-triangle-inequality-lt-sep
∀e:BasicGeometry. ∀a,b,c,d,g,h:Point.
(|ab| < |gh| + |cd| ⇒ |cd| < |ab| + |gh| ⇒ |gh| < |cd| + |ab| ⇒ ((a ≠ b ∧ g ≠ h) ∧ c ≠ d))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
false: False
Lemmas referenced :
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-add-length-lt-sep,
geo-sep-iff-or-lt,
geo-length_wf1,
geo-add-length_wf1,
geo-sep-or,
geo-sep_wf,
geo-lt_transitivity,
geo-le_weakening-lt,
squash_wf,
true_wf,
geo-length-type_wf,
geo-add-length-zero2,
subtype_rel_self,
iff_weakening_equal,
geo-add-length-cancel-left-lt,
geo-lt-sep,
geo-add-length-comm,
geo-lt-null-segment
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
setElimination,
rename,
because_Cache,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
inlFormation_alt,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
voidElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d,g,h:Point.
(|ab| < |gh| + |cd| {}\mRightarrow{} |cd| < |ab| + |gh| {}\mRightarrow{} |gh| < |cd| + |ab| {}\mRightarrow{} ((a \mneq{} b \mwedge{} g \mneq{} h) \mwedge{} c \mneq{} d))
Date html generated:
2019_10_16-PM-01_39_12
Last ObjectModification:
2019_02_19-PM-01_06_35
Theory : euclidean!plane!geometry
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