Nuprl Lemma : old-proof-of-real-continuity
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ. real-cont(f;a;b) supposing real-fun(f;a;b) supposing a ≤ b
Proof
Definitions occuring in Statement :
real-cont: real-cont(f;a;b),
real-fun: real-fun(f;a;b),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
real-fun: real-fun(f;a;b),
implies: P ⇒ Q,
rfun: I ⟶ℝ,
prop: ℙ,
real-cont: real-cont(f;a;b),
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s1;s2],
i-member: r ∈ I,
rccint: [l, u],
real: ℝ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
real-sfun: real-sfun(f;a;b),
sq_stable: SqStable(P),
cand: A c∧ B,
squash: ↓T,
rneq: x ≠ y,
guard: {T},
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
less_than: a < b,
less_than': less_than'(a;b),
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
rational-approx: (x within 1/n),
rge: x ≥ y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. real-cont(f;a;b) supposing real-fun(f;a;b) supposing a \mleq{} b
Date html generated:
2020_05_20-PM-00_09_51
Last ObjectModification:
2019_12_14-PM-03_09_41
Theory : reals
Home
Index