Nuprl Lemma : continuous-implies-functional
∀I:Interval. ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ (∀a,b:{x:ℝ| x ∈ I} . ((a = b) ⇒ (f[a] = f[b]))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ 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],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_apply: x[s],
rfun: I ⟶ℝ,
uimplies: b supposing a,
not: ¬A,
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t
Lemmas referenced :
continuous-rneq,
not-rneq,
req_inversion,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity,
rneq_wf,
req_wf,
real_wf,
i-member_wf,
continuous_wf,
rfun_wf,
interval_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
applyEquality,
sqequalRule,
independent_isectElimination,
unionElimination,
setElimination,
rename,
because_Cache,
voidElimination,
setEquality,
lambdaEquality
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(f[x] continuous for x \mmember{} I {}\mRightarrow{} (\mforall{}a,b:\{x:\mBbbR{}| x \mmember{} I\} . ((a = b) {}\mRightarrow{} (f[a] = f[b]))))
Date html generated:
2016_05_18-AM-09_09_09
Last ObjectModification:
2015_12_27-PM-11_30_27
Theory : reals
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