Nuprl Lemma : continuous-add
∀[I:Interval]. ∀[f,g:I ⟶ℝ].
(f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ I ⇒ f[x] + g[x] continuous for x ∈ I)
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
radd: a + b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
all: ∀x:A. B[x],
member: t ∈ T,
nat_plus: ℕ+,
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,
and: P ∧ Q,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rless: x < y,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
rneq: x ≠ y,
rev_implies: P ⇐ Q,
r-ap: f(x),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2
Latex:
\mforall{}[I:Interval]. \mforall{}[f,g:I {}\mrightarrow{}\mBbbR{}].
(f[x] continuous for x \mmember{} I {}\mRightarrow{} g[x] continuous for x \mmember{} I {}\mRightarrow{} f[x] + g[x] continuous for x \mmember{} I)
Date html generated:
2020_05_20-PM-00_14_01
Last ObjectModification:
2019_12_14-PM-03_10_47
Theory : reals
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