Nuprl Lemma : geo-colinear-out-cases
∀[e:BasicGeometry-]. ∀[a,b,c:Point]. ∀X:ℙ. (Stable{X} ⇒ Colinear(a;b;c) ⇒ (out(a bc) ⇒ X) ⇒ (c_a_b ⇒ X) ⇒ X)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry-: BasicGeometry-,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-point: Point,
stable: Stable{P},
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
basic-geometry-: BasicGeometry-,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False
Lemmas referenced :
geo-simple-colinear-cases,
geo-between_wf,
geo-out_wf,
euclidean-plane-subtype-basic,
basic-geometry--subtype,
subtype_rel_transitivity,
basic-geometry-_wf,
euclidean-plane_wf,
basic-geometry_wf,
geo-colinear_wf,
stable_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
euclidean-plane-structure_wf,
geo-primitives_wf,
false_wf,
or_wf,
not_wf,
geo-colinear-not-out,
geo-between-symmetry,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
applyEquality,
because_Cache,
sqequalRule,
instantiate,
independent_isectElimination,
universeEquality,
productElimination,
unionElimination,
voidElimination
Latex:
\mforall{}[e:BasicGeometry-]. \mforall{}[a,b,c:Point].
\mforall{}X:\mBbbP{}. (Stable\{X\} {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} (out(a bc) {}\mRightarrow{} X) {}\mRightarrow{} (c\_a\_b {}\mRightarrow{} X) {}\mRightarrow{} X)
Date html generated:
2018_05_22-AM-11_58_35
Last ObjectModification:
2018_04_19-PM-01_49_48
Theory : euclidean!plane!geometry
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