Nuprl Lemma : rleq-range_sup-2
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀[c:ℝ]. c ≤ sup{f[x] | x ∈ I} supposing ∃x:ℝ. ((x ∈ I) ∧ (c ≤ f[x]))
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],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T}
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}.
\mforall{}[c:\mBbbR{}]. c \mleq{} sup\{f[x] | x \mmember{} I\} supposing \mexists{}x:\mBbbR{}. ((x \mmember{} I) \mwedge{} (c \mleq{} f[x]))
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_48
Last ObjectModification:
2020_01_03-PM-01_37_31
Theory : reals
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