### Nuprl Lemma : subtype-l_all

`∀[T:Type]. ∀[L:T List]. ∀[P,Q:{x:T| (x ∈ L)}  ⟶ ℙ].`
`  (∀x∈L.P[x]) ⊆r (∀x∈L.Q[x]) supposing ∀x:T. ((x ∈ L) `` (P[x] ⊆r Q[x]))`

Proof

Definitions occuring in Statement :  l_all: `(∀x∈L.P[x])` l_member: `(x ∈ l)` list: `T List` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` l_all: `(∀x∈L.P[x])` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T`
Lemmas referenced :  subtype_rel_wf l_member_wf all_wf select_member int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le int_seg_properties length_wf int_seg_wf select_wf subtype_rel_dep_function
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaEquality sqequalHypSubstitution hypothesisEquality applyEquality lemma_by_obid isectElimination thin because_Cache sqequalRule independent_isectElimination hypothesis natural_numberEquality lambdaFormation dependent_functionElimination setElimination rename productElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination axiomEquality cumulativity functionEquality dependent_set_memberEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P,Q:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbP{}].
(\mforall{}x\mmember{}L.P[x])  \msubseteq{}r  (\mforall{}x\mmember{}L.Q[x])  supposing  \mforall{}x:T.  ((x  \mmember{}  L)  {}\mRightarrow{}  (P[x]  \msubseteq{}r  Q[x]))

Date html generated: 2016_05_14-AM-07_47_30
Last ObjectModification: 2016_01_15-AM-08_34_32

Theory : list_1

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