Nuprl Lemma : l_all_map
∀[A,B:Type]. ∀f:A ⟶ B. ∀L:A List. ∀[P:B ⟶ ℙ]. ((∀x∈map(f;L).P[x]) ⇐⇒ (∀x∈L.P[f x]))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
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,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
l_member_wf,
equal_wf,
all_wf,
exists_wf,
member_map,
map_wf,
l_all_iff,
l_all_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
applyEquality,
hypothesisEquality,
independent_functionElimination,
dependent_pairFormation,
because_Cache,
productEquality,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
productElimination,
cumulativity,
setElimination,
rename,
setEquality,
universeEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. \mforall{}L:A List. \mforall{}[P:B {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x\mmember{}map(f;L).P[x]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x\mmember{}L.P[f x]))
Date html generated:
2019_06_20-PM-00_41_45
Last ObjectModification:
2018_09_18-PM-02_28_25
Theory : list_0
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