Nuprl Lemma : infn_functionality
∀n:ℕ. ∀I:{I:Interval| icompact(I)} . ∀f,g:I^n ⟶ ℝ.
((∀x,y:I^n. (req-vec(n;x;y) ⇒ ((f x) = (f y)))) ⇒ (∀x:I^n. ((f x) = (g x))) ⇒ ((infn(n;I) f) = (infn(n;I) g)))
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,
req: x = y,
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
interval-vec: I^n,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
guard: {T},
so_apply: x[s],
infn: infn(n;I),
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
nat: ℕ,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
req-vec: req-vec(n;x;y),
real-vec-extend: a++z,
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}n:\mBbbN{}. \mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}f,g:I\^{}n {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:I\^{}n. (req-vec(n;x;y) {}\mRightarrow{} ((f x) = (f y))))
{}\mRightarrow{} (\mforall{}x:I\^{}n. ((f x) = (g x)))
{}\mRightarrow{} ((infn(n;I) f) = (infn(n;I) g)))
Date html generated:
2020_05_20-PM-00_39_01
Last ObjectModification:
2020_01_06-PM-09_55_17
Theory : reals
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