Nuprl Lemma : geo-add-length-lt-sep2
∀e:BasicGeometry. ∀a,b,c,d,g,h:Point. (|ab| < |cd| + |gh| ⇒ |ab| ≠ |cd| + |gh|)
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
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
geo-lt-iff-strict-between-points,
geo-length_wf1,
geo-mk-seg_wf,
geo-add-length_wf1,
geo-lt_wf,
geo-length_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
independent_functionElimination,
universeIsType,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d,g,h:Point. (|ab| < |cd| + |gh| {}\mRightarrow{} |ab| \mneq{} |cd| + |gh|)
Date html generated:
2019_10_16-PM-01_38_45
Last ObjectModification:
2019_02_27-PM-03_44_55
Theory : euclidean!plane!geometry
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