Nuprl Lemma : infn-nonneg
∀[I:{I:Interval| icompact(I)} ]
∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} .
r0 ≤ (infn(n;I) f) supposing ∀a:I^n. (r0 ≤ (f a))
Proof
Definitions occuring in Statement :
infn: infn(n;I),
interval-vec: I^n,
req-vec: req-vec(n;x;y),
icompact: icompact(I),
interval: Interval,
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
interval-vec: I^n
Lemmas referenced :
rleq-infn,
le_witness_for_triv,
interval-vec_wf,
rleq_wf,
int-to-real_wf,
real_wf,
req-vec_wf,
req_wf,
istype-nat,
interval_wf,
icompact_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
hypothesis,
sqequalRule,
lambdaEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
universeIsType,
setElimination,
rename,
natural_numberEquality,
applyEquality,
setIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[I:\{I:Interval| icompact(I)\} ]
\mforall{}n:\mBbbN{}. \mforall{}f:\{f:I\^{}n {}\mrightarrow{} \mBbbR{}| \mforall{}a,b:I\^{}n. (req-vec(n;a;b) {}\mRightarrow{} ((f a) = (f b)))\} .
r0 \mleq{} (infn(n;I) f) supposing \mforall{}a:I\^{}n. (r0 \mleq{} (f a))
Date html generated:
2019_10_30-AM-08_25_47
Last ObjectModification:
2019_05_28-PM-05_29_31
Theory : reals
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