Nuprl Lemma : product-map
∀[A,B,C:Type].
∀as:A List. ∀bs:B List. ∀F:A ⟶ B ⟶ C. ∃cs:C List. ∀c:C. ((c ∈ cs) ⇐⇒ (∃a∈as. (∃b∈bs. c = (F a b) ∈ C)))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
product-map_wf,
exists_wf,
l_member_wf,
equal_wf,
member-product-map,
l_exists_iff,
l_exists_wf,
iff_wf,
all_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
independent_pairFormation,
sqequalRule,
lambdaEquality,
productEquality,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
existsFunctionality,
setElimination,
rename,
setEquality,
andLevelFunctionality,
levelHypothesis,
existsLevelFunctionality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B,C:Type].
\mforall{}as:A List. \mforall{}bs:B List. \mforall{}F:A {}\mrightarrow{} B {}\mrightarrow{} C.
\mexists{}cs:C List. \mforall{}c:C. ((c \mmember{} cs) \mLeftarrow{}{}\mRightarrow{} (\mexists{}a\mmember{}as. (\mexists{}b\mmember{}bs. c = (F a b))))
Date html generated:
2018_05_22-AM-00_19_50
Last ObjectModification:
2017_07_26-PM-06_54_24
Theory : rationals
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