Nuprl Lemma : geo-out-strict_functionality
∀e:BasicGeometry. ∀p,a,b,p',a',b':Point.
(p ≡ p' ⇒ a ≡ a' ⇒ b ≡ b' ⇒ {geo-out-strict(e;p;a;b) ⇐⇒ geo-out-strict(e;p';a';b')})
Proof
Definitions occuring in Statement :
geo-out-strict: geo-out-strict(e;p;a;b),
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
geo-out-strict: geo-out-strict(e;p;a;b),
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A
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-eq_wf,
geo-out-strict_wf,
geo-eq_inversion,
geo-strict-between_functionality,
geo-strict-between_wf
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
levelHypothesis,
impliesLevelFunctionality,
andLevelFunctionality,
independent_functionElimination,
dependent_functionElimination,
productElimination,
impliesFunctionality,
addLevel
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,a,b,p',a',b':Point.
(p \mequiv{} p' {}\mRightarrow{} a \mequiv{} a' {}\mRightarrow{} b \mequiv{} b' {}\mRightarrow{} \{geo-out-strict(e;p;a;b) \mLeftarrow{}{}\mRightarrow{} geo-out-strict(e;p';a';b')\})
Date html generated:
2017_10_02-PM-06_25_31
Last ObjectModification:
2017_08_05-PM-04_19_05
Theory : euclidean!plane!geometry
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