Nuprl Lemma : i-member-convex3
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (∀n:ℕ+. ∀i:ℕ. (((i + 1) = n ∈ ℤ) ⇒ ((i * a + b)/n ∈ I))))
Proof
Definitions occuring in Statement :
i-member: r ∈ I,
interval: Interval,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
radd: a + b,
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
nat: ℕ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
i-member-convex2,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
int-rmul-one,
set_subtype_base,
le_wf,
int_subtype_base,
less_than_wf,
istype-nat,
nat_plus_wf,
i-member_wf,
real_wf,
interval_wf,
int-rdiv_wf,
nat_plus_inc_int_nzero,
radd_wf,
int-rmul_wf,
i-member_functionality,
int-rdiv_functionality,
radd_functionality,
req_weakening
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
isectElimination,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
equalitySymmetry,
inhabitedIsType,
because_Cache,
equalityTransitivity,
productElimination
Latex:
\mforall{}I:Interval. \mforall{}a,b:\mBbbR{}. ((a \mmember{} I) {}\mRightarrow{} (b \mmember{} I) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}i:\mBbbN{}. (((i + 1) = n) {}\mRightarrow{} ((i * a + b)/n \mmember{} I))))
Date html generated:
2019_10_29-AM-10_47_28
Last ObjectModification:
2019_01_08-PM-10_02_52
Theory : reals
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