Nuprl Lemma : geo-lt-iff-strict-between
∀g:EuclideanPlane. ∀s1,s2:geo-segment(g). (|s1| < |s2| ⇐⇒ X_|s1|_|s2| ∧ |s1| ≠ |s2|)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-segment: geo-segment(e),
geo-X: X,
euclidean-plane: EuclideanPlane,
geo-between: a_b_c,
geo-sep: a ≠ b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
euclidean-plane: EuclideanPlane
Lemmas referenced :
geo-lt-iff-strict-between-points,
geo-length_wf1,
geo-le-iff-between,
geo-lt_wf,
geo-length_wf,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-X_wf,
geo-sep_wf,
geo-segment_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
sqequalRule,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
promote_hyp,
universeIsType,
productIsType,
applyEquality,
instantiate,
independent_isectElimination,
setElimination,
rename,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache
Latex:
\mforall{}g:EuclideanPlane. \mforall{}s1,s2:geo-segment(g). (|s1| < |s2| \mLeftarrow{}{}\mRightarrow{} X\_|s1|\_|s2| \mwedge{} |s1| \mneq{} |s2|)
Date html generated:
2019_10_16-PM-01_34_40
Last ObjectModification:
2018_10_03-AM-11_10_37
Theory : euclidean!plane!geometry
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