Nuprl Lemma : l_all_iff
∀[T:Type]. ∀L:T List. ∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. ((∀x∈L.P[x]) ⇐⇒ ∀x:T. ((x ∈ L) ⇒ P[x]))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
l_all: (∀x∈L.P[x]),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
cand: A c∧ B,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
l_member_wf,
l_all_wf,
all_wf,
list_wf,
decidable__lt,
length_wf,
false_wf,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
lelt_wf,
select_wf,
list-subtype,
sq_stable__le,
nat_wf,
select_member,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
setElimination,
rename,
dependent_set_memberEquality,
functionEquality,
because_Cache,
universeEquality,
productElimination,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
natural_numberEquality,
isect_memberEquality,
voidEquality,
intEquality,
addLevel,
levelHypothesis,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x\mmember{}L.P[x]) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} P[x]))
Date html generated:
2016_10_21-AM-09_48_59
Last ObjectModification:
2016_07_12-AM-05_08_38
Theory : list_0
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