Nuprl Lemma : int_le_to_int_upper
∀i:ℤ. ∀[A:{i...} ⟶ ℙ]. ({∀j:ℤ. A[j] supposing i ≤ j} ⇐⇒ {∀j:{i...}. A[j]})
Proof
Definitions occuring in Statement :
int_upper: {i...},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
int_upper: {i...},
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
le: A ≤ B,
not: ¬A,
false: False
Lemmas referenced :
less_than'_wf,
le_wf,
isect_wf,
all_wf,
int_upper_wf,
sq_stable__le
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
independent_isectElimination,
lemma_by_obid,
isectElimination,
independent_functionElimination,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
intEquality,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
because_Cache,
productElimination,
independent_pairEquality,
voidElimination,
axiomEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}[A:\{i...\} {}\mrightarrow{} \mBbbP{}]. (\{\mforall{}j:\mBbbZ{}. A[j] supposing i \mleq{} j\} \mLeftarrow{}{}\mRightarrow{} \{\mforall{}j:\{i...\}. A[j]\})
Date html generated:
2016_05_13-PM-04_02_43
Last ObjectModification:
2016_01_14-PM-07_24_31
Theory : int_1
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