Nuprl Lemma : geo-le-pt-transitivity
∀e:BasicGeometry. ∀a,b,c,d,c',d':Point. (a ≠ b ⇒ ab≤cd ⇒ cd≤c'd' ⇒ ab≤c'd')
Proof
Definitions occuring in Statement :
geo-le-pt: ab≤cd,
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
guard: {T},
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
prop: ℙ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-le-pt: ab≤cd,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
cand: A c∧ B
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-le-pt_wf,
geo-congruent-sep,
geo-congruent-symmetry,
geo-congruent-between-exists,
geo-congruent_wf,
geo-between_wf,
geo-congruent-iff-length,
geo-between-exchange4,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry
Rules used in proof :
sqequalRule,
instantiate,
applyEquality,
rename,
setElimination,
hypothesis,
independent_isectElimination,
isectElimination,
because_Cache,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
equalityTransitivity,
independent_pairFormation,
dependent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d,c',d':Point. (a \mneq{} b {}\mRightarrow{} ab\mleq{}cd {}\mRightarrow{} cd\mleq{}c'd' {}\mRightarrow{} ab\mleq{}c'd')
Date html generated:
2017_10_02-PM-06_46_27
Last ObjectModification:
2017_08_05-PM-04_51_21
Theory : euclidean!plane!geometry
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