Nuprl Lemma : geo-same-side-iff
∀[e:BasicGeometry]. ∀[A,B,P,Q:Point].
(P,Q-AB ⇐⇒ ((¬Colinear(A;B;P)) ∧ (¬Colinear(A;B;Q))) ∧ (¬P leftof AB ⇐⇒ ¬Q leftof AB))
Proof
Definitions occuring in Statement :
geo-same-side: A,B-PQ,
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-point: Point,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
geo-same-side: A,B-PQ,
all: ∀x:A. B[x],
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
geo-lsep: a # bc,
or: P ∨ Q,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
select: L[n],
cons: [a / b],
subtract: n - m,
stable: Stable{P},
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
geo-colinear: Colinear(a;b;c)
Lemmas referenced :
geo-between-trivial2,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-between-trivial,
geo-left_wf,
istype-void,
geo-same-side_wf,
geo-point_wf,
stable__false,
false_wf,
or_wf,
geo-lsep_wf,
not_wf,
not-lsep-iff-colinear,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-SS_wf,
geo-between-symmetry,
geo-between_wf,
left-between,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
lsep-all-sym2,
lsep-all-sym,
geo-between-implies-colinear
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
thin,
sqequalHypSubstitution,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
extract_by_obid,
isectElimination,
voidElimination,
universeIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
functionIsType,
productElimination,
productIsType,
independent_pairEquality,
lambdaEquality_alt,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionEquality,
unionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
unionIsType,
setElimination,
rename,
equalityIstype,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[A,B,P,Q:Point].
(P,Q-AB \mLeftarrow{}{}\mRightarrow{} ((\mneg{}Colinear(A;B;P)) \mwedge{} (\mneg{}Colinear(A;B;Q))) \mwedge{} (\mneg{}P leftof AB \mLeftarrow{}{}\mRightarrow{} \mneg{}Q leftof AB))
Date html generated:
2019_10_16-PM-01_20_50
Last ObjectModification:
2018_12_11-PM-00_15_45
Theory : euclidean!plane!geometry
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