Nuprl Lemma : eu-colinear-is-colinear-set
∀e:EuclideanPlane. ∀A,B,C:Point. (Colinear(A;B;C) ⇒ eu-colinear-set(e;[A; B; C]))
Proof
Definitions occuring in Statement :
eu-colinear-set: eu-colinear-set(e;L),
euclidean-plane: EuclideanPlane,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
cons: [a / b],
nil: [],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
eu-colinear-set: eu-colinear-set(e;L),
l_all: (∀x∈L.P[x]),
member: t ∈ T,
top: Top,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
select: L[n],
cons: [a / b],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ,
not: ¬A,
false: False,
subtract: n - m,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
less_than': less_than'(a;b),
lelt: i ≤ j < k,
stable: Stable{P}
Lemmas referenced :
length_of_cons_lemma,
length_of_nil_lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
l_all_iff,
cons_wf,
eu-point_wf,
nil_wf,
l_member_wf,
l_all_wf2,
not_wf,
equal_wf,
eu-colinear_wf,
eu-colinear-def,
member_wf,
eu-between_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_subtype,
false_wf,
int_seg_cases,
eu-colinear-swap,
eu-colinear-cycle,
eu-colinear-permute,
int_seg_wf,
length_wf,
euclidean-plane_wf,
stable__colinear,
or_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
functionEquality,
setEquality,
productElimination,
independent_pairFormation,
productEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
hyp_replacement,
Error :applyLambdaEquality,
hypothesis_subsumption,
addEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A,B,C:Point. (Colinear(A;B;C) {}\mRightarrow{} eu-colinear-set(e;[A; B; C]))
Date html generated:
2016_10_26-AM-07_43_33
Last ObjectModification:
2016_07_12-AM-08_11_07
Theory : euclidean!geometry
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