Nuprl Lemma : i-member-convex2
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (∀n:ℕ+. ∀i,j:ℕ. (((i + j) = n ∈ ℤ) ⇒ ((i * a + j * 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,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
nat_plus: ℕ+,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
true: True
Latex:
\mforall{}I:Interval. \mforall{}a,b:\mBbbR{}.
((a \mmember{} I) {}\mRightarrow{} (b \mmember{} I) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}i,j:\mBbbN{}. (((i + j) = n) {}\mRightarrow{} ((i * a + j * b)/n \mmember{} I))))
Date html generated:
2020_05_20-AM-11_34_43
Last ObjectModification:
2020_01_02-PM-02_00_25
Theory : reals
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