Nuprl Lemma : range_sup-bound
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀[c:ℝ]. sup{f[x] | x ∈ I} ≤ c supposing ∀x:ℝ. ((x ∈ I) ⇒ (f[x] ≤ c))
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Proof
Definitions occuring in Statement :
range_sup: sup{f[x] | x ∈ I},
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
label: ...$L... t,
rfun: I ⟶ℝ,
uiff: uiff(P;Q),
sup: sup(A) = b,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
rrange: f[x](x∈I),
rset-member: x ∈ A,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
iff: P ⇐⇒ Q
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}.
\mforall{}[c:\mBbbR{}]. sup\{f[x] | x \mmember{} I\} \mleq{} c supposing \mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} (f[x] \mleq{} c))
supposing \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[x] = f[y]))
Date html generated:
2020_05_20-PM-00_16_03
Last ObjectModification:
2020_01_03-PM-02_33_33
Theory : reals
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