Nuprl Lemma : geo-strict-between_functionality
∀e:BasicGeometry. ∀a1,a2,b1,b2,c1,c2:Point. (a1 ≡ a2 ⇒ b1 ≡ b2 ⇒ c1 ≡ c2 ⇒ (a1-b1-c1 ⇐⇒ a2-b2-c2))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
geo-strict-between: a-b-c,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
cand: A c∧ B
Lemmas referenced :
basic-geometry_wf,
geo-point_wf,
geo-eq_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-strict-between_wf,
geo-sep_wf,
geo-between_wf,
geo-sep_functionality,
geo-between_functionality,
geo-eq_inversion
Rules used in proof :
hypothesis,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
andLevelFunctionality,
promote_hyp,
levelHypothesis,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
productElimination,
addLevel
Latex:
\mforall{}e:BasicGeometry. \mforall{}a1,a2,b1,b2,c1,c2:Point.
(a1 \mequiv{} a2 {}\mRightarrow{} b1 \mequiv{} b2 {}\mRightarrow{} c1 \mequiv{} c2 {}\mRightarrow{} (a1-b1-c1 \mLeftarrow{}{}\mRightarrow{} a2-b2-c2))
Date html generated:
2017_10_02-PM-04_42_54
Last ObjectModification:
2017_08_05-AM-08_42_20
Theory : euclidean!plane!geometry
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