Nuprl Lemma : continuous-sub
∀[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,
rsub: x - y,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q
Definitions unfolded in proof :
rsub: x - y,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
label: ...$L... t
Lemmas referenced :
continuous-add,
real_wf,
i-member_wf,
rminus_wf,
continuous-minus,
continuous_wf,
rfun_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
setEquality,
hypothesis,
because_Cache,
independent_functionElimination
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:
2016_05_18-AM-09_11_47
Last ObjectModification:
2015_12_27-PM-11_28_17
Theory : reals
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