### Nuprl Lemma : member-product-map

`∀[A,B,C:Type].`
`  ∀F:A ⟶ B ⟶ C. ∀as:A List. ∀bs:B List. ∀c:C.`
`    ((c ∈ product-map(F;as;bs)) `⇐⇒` ∃a:A. ((a ∈ as) ∧ (∃b:B. ((b ∈ bs) ∧ (c = (F a b) ∈ C)))))`

Proof

Definitions occuring in Statement :  product-map: `product-map(F;as;bs)` 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` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  product-map: `product-map(F;as;bs)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` so_apply: `x[s]` exists: `∃x:A. B[x]` implies: `P `` Q` top: `Top` concat: `concat(ll)` iff: `P `⇐⇒` Q` false: `False` rev_implies: `P `` Q` or: `P ∨ Q` cand: `A c∧ B` guard: `{T}`
Lemmas referenced :  list_induction all_wf list_wf iff_wf l_member_wf concat_wf map_wf exists_wf equal_wf map_nil_lemma reduce_nil_lemma false_wf nil_member nil_wf map_cons_lemma or_wf member-map member_append append_wf concat-cons cons_member cons_wf and_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality lambdaEquality cumulativity hypothesis because_Cache applyEquality functionExtensionality productEquality independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation productElimination addLevel allFunctionality impliesFunctionality existsFunctionality andLevelFunctionality existsLevelFunctionality rename orFunctionality functionEquality universeEquality levelHypothesis promote_hyp unionElimination dependent_pairFormation inlFormation inrFormation equalitySymmetry dependent_set_memberEquality applyLambdaEquality setElimination equalityTransitivity

Latex:
\mforall{}[A,B,C:Type].
\mforall{}F:A  {}\mrightarrow{}  B  {}\mrightarrow{}  C.  \mforall{}as:A  List.  \mforall{}bs:B  List.  \mforall{}c:C.
((c  \mmember{}  product-map(F;as;bs))  \mLeftarrow{}{}\mRightarrow{}  \mexists{}a:A.  ((a  \mmember{}  as)  \mwedge{}  (\mexists{}b:B.  ((b  \mmember{}  bs)  \mwedge{}  (c  =  (F  a  b))))))

Date html generated: 2017_04_14-AM-09_27_16
Last ObjectModification: 2017_02_27-PM-04_01_01

Theory : list_1

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