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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