Nuprl Lemma : closures-meet
∀[P,Q:ℝ ⟶ ℙ].
((∃a,b:ℝ. ((P a) ∧ (Q b) ∧ (a ≤ b)))
⇒ (∃c:ℝ
(((r0 ≤ c) ∧ (c < r1))
∧ (∀a,b:ℝ.
(((P a) ∧ (Q b) ∧ (a ≤ b))
⇒ (∃a',b':ℝ. ((P a') ∧ (Q b') ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))))))))
⇒ (∃y:ℝ. (y ∈ closure(P) ∧ y ∈ closure(Q))))
Proof
Definitions occuring in Statement :
member-closure: y ∈ closure(A),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
cand: A c∧ B,
spreadn: spread3,
pi1: fst(t),
nat: ℕ,
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
rless: x < y,
sq_exists: ∃x:A [B[x]],
false: False,
nat_plus: ℕ+,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
pi2: snd(t),
rbetween: x≤y≤z,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
so_lambda: λ2x.t[x],
rmul: a * b,
rsub: x - y,
radd: a + b,
accelerate: accelerate(k;f),
rnexp: x^k1,
member-closure: y ∈ closure(A)
Latex:
\mforall{}[P,Q:\mBbbR{} {}\mrightarrow{} \mBbbP{}].
((\mexists{}a,b:\mBbbR{}. ((P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} b)))
{}\mRightarrow{} (\mexists{}c:\mBbbR{}
(((r0 \mleq{} c) \mwedge{} (c < r1))
\mwedge{} (\mforall{}a,b:\mBbbR{}.
(((P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} b))
{}\mRightarrow{} (\mexists{}a',b':\mBbbR{}
((P a')
\mwedge{} (Q b')
\mwedge{} (a \mleq{} a')
\mwedge{} (a' \mleq{} b')
\mwedge{} (b' \mleq{} b)
\mwedge{} ((b' - a') \mleq{} ((b - a) * c))))))))
{}\mRightarrow{} (\mexists{}y:\mBbbR{}. (y \mmember{} closure(P) \mwedge{} y \mmember{} closure(Q))))
Date html generated:
2020_05_20-AM-11_29_04
Last ObjectModification:
2019_12_14-PM-04_50_56
Theory : reals
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