Nuprl Lemma : function-proper-continuous
∀I:Interval. ∀f:I ⟶ℝ.
((iproper(I) ⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[x] = f[y])))) ⇒ f[x] (proper)continuous for x ∈ I)
Proof
Definitions occuring in Statement :
proper-continuous: f[x] (proper)continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
proper-continuous: f[x] (proper)continuous for x ∈ I,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
cand: A c∧ B,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
so_apply: x[s],
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
subinterval: I ⊆ J ,
continuous: f[x] continuous for x ∈ I
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
((iproper(I) {}\mRightarrow{} (\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[x] = f[y]))))
{}\mRightarrow{} f[x] (proper)continuous for x \mmember{} I)
Date html generated:
2020_05_20-PM-00_10_14
Last ObjectModification:
2019_12_18-AM-10_01_08
Theory : reals
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