Nuprl Lemma : l_all_functionality
∀[T:Type]. ∀L:T List. ∀P,Q:T ⟶ ℙ. ((∀x:T. ((x ∈ L) ⇒ (P[x] ⇐⇒ Q[x]))) ⇒ {(∀x∈L.P[x]) ⇐⇒ (∀x∈L.Q[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,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
l_all: (∀x∈L.P[x]),
member: t ∈ T,
int_seg: {i..j-},
uimplies: b supposing a,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
list_wf,
iff_wf,
all_wf,
l_member_wf,
l_all_wf,
length_wf,
int_seg_wf,
select_member,
sq_stable__le,
select_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
lemma_by_obid,
isectElimination,
cumulativity,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
independent_functionElimination,
introduction,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
applyEquality,
setEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P,Q:T {}\mrightarrow{} \mBbbP{}. ((\mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} (P[x] \mLeftarrow{}{}\mRightarrow{} Q[x]))) {}\mRightarrow{} \{(\mforall{}x\mmember{}L.P[x]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x\mmember{}L.Q[x])\})
Date html generated:
2016_05_14-AM-06_40_39
Last ObjectModification:
2016_01_14-PM-08_19_53
Theory : list_0
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