Nuprl Lemma : cover-seq_wf
∀[A,B:ℝ ⟶ ℙ]. ∀[d:r:ℝ ⟶ (A[r] + B[r])]. ∀[a,b:ℝ]. ∀[n:ℕ]. (cover-seq(d;a;b;n) ∈ ℝ × ℝ)
Proof
Definitions occuring in Statement :
cover-seq: cover-seq(d;a;b;n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
union: left + right
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cover-seq: cover-seq(d;a;b;n),
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ
Lemmas referenced :
primrec_wf,
real_wf,
rdiv_wf,
radd_wf,
int-to-real_wf,
rless-int,
rless_wf,
equal_wf,
int_seg_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesis,
because_Cache,
hypothesisEquality,
independent_pairEquality,
lambdaEquality,
productElimination,
applyEquality,
functionExtensionality,
natural_numberEquality,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
unionEquality,
lambdaFormation,
unionElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
axiomEquality,
isect_memberEquality,
functionEquality,
universeEquality,
cumulativity
Latex:
\mforall{}[A,B:\mBbbR{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:r:\mBbbR{} {}\mrightarrow{} (A[r] + B[r])]. \mforall{}[a,b:\mBbbR{}]. \mforall{}[n:\mBbbN{}]. (cover-seq(d;a;b;n) \mmember{} \mBbbR{} \mtimes{} \mBbbR{})
Date html generated:
2017_10_03-AM-10_02_48
Last ObjectModification:
2017_07_06-AM-10_51_53
Theory : reals
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