### Nuprl Lemma : prod-metric_wf

`∀[k:ℕ]. ∀[X:ℕk ⟶ Type]. ∀[d:i:ℕk ⟶ metric(X[i])].  (prod-metric(k;d) ∈ metric(i:ℕk ⟶ X[i]))`

Proof

Definitions occuring in Statement :  prod-metric: `prod-metric(k;d)` metric: `metric(X)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  prod-metric: `prod-metric(k;d)` uall: `∀[x:A]. B[x]` member: `t ∈ T` metric: `metric(X)` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` prop: `ℙ` cand: `A c∧ B` le: `A ≤ B` less_than: `a < b` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` pointwise-rleq: `x[k] ≤ y[k] for k ∈ [n,m]` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  rsum_wf subtract_wf mdist_wf subtract-add-cancel nat_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_wf istype-le istype-less_than int_seg_wf rsum-of-nonneg-zero-iff mdist-nonneg int_seg_properties decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma itermAdd_wf itermSubtract_wf int_term_value_add_lemma int_term_value_subtract_lemma mdist-same rleq_wf radd_wf req_wf int-to-real_wf metric_wf istype-universe istype-nat rsum_functionality_wrt_rleq mdist-triangle-inequality1 rleq_functionality req_weakening req_inversion rsum_linearity1
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation_alt introduction cut dependent_set_memberEquality_alt lambdaEquality_alt extract_by_obid sqequalHypSubstitution isectElimination thin closedConclusion natural_numberEquality setElimination rename because_Cache hypothesis applyEquality hypothesisEquality productElimination independent_pairFormation dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination universeIsType productIsType addEquality inhabitedIsType functionIsType lambdaFormation_alt imageElimination axiomEquality equalityTransitivity equalitySymmetry isectIsTypeImplies instantiate universeEquality

Latex:
\mforall{}[k:\mBbbN{}].  \mforall{}[X:\mBbbN{}k  {}\mrightarrow{}  Type].  \mforall{}[d:i:\mBbbN{}k  {}\mrightarrow{}  metric(X[i])].    (prod-metric(k;d)  \mmember{}  metric(i:\mBbbN{}k  {}\mrightarrow{}  X[i]))

Date html generated: 2019_10_29-AM-11_09_34
Last ObjectModification: 2019_10_02-AM-09_50_39

Theory : reals

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