Nuprl Lemma : range_sup_functionality2
∀I,J:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀g:{x:ℝ| x ∈ J} ⟶ ℝ
(sup{f[x] | x ∈ I} = sup{g[x] | x ∈ J}) supposing
(((∀x:{x:ℝ| x ∈ I} . ∃y:{x:ℝ| x ∈ J} . (f[x] = g[y])) ∧ (∀x:{x:ℝ| x ∈ J} . ∃y:{x:ℝ| x ∈ I} . (f[y] = g[x]))) and
(∀x,y:{x:ℝ| x ∈ J} . ((x = y) ⇒ (g[x] = g[y]))))
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,
req: x = y,
real: ℝ,
uimplies: b supposing a,
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,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Latex:
\mforall{}I,J:\{I:Interval| icompact(I)\} . \mforall{}f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}.
\mforall{}g:\{x:\mBbbR{}| x \mmember{} J\} {}\mrightarrow{} \mBbbR{}
(sup\{f[x] | x \mmember{} I\} = sup\{g[x] | x \mmember{} J\}) supposing
(((\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . \mexists{}y:\{x:\mBbbR{}| x \mmember{} J\} . (f[x] = g[y]))
\mwedge{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} J\} . \mexists{}y:\{x:\mBbbR{}| x \mmember{} I\} . (f[y] = g[x]))) and
(\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} J\} . ((x = y) {}\mRightarrow{} (g[x] = g[y]))))
supposing \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[x] = f[y]))
Date html generated:
2020_05_20-PM-00_19_03
Last ObjectModification:
2020_01_03-PM-03_19_40
Theory : reals
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