### Nuprl Lemma : strong-continuity2-implies-weak-skolem-cantor-nat

`∀F:(ℕ ⟶ 𝔹) ⟶ ℕ. ⇃(∃M:(ℕ ⟶ 𝔹) ⟶ ℕ. ∀f,g:ℕ ⟶ 𝔹.  ((f = g ∈ (ℕM f ⟶ 𝔹)) `` ((F f) = (F g) ∈ ℕ)))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` implies: `P `` Q` exists: `∃x:A. B[x]` prop: `ℙ` uall: `∀[x:A]. B[x]` nat: `ℕ` so_lambda: `λ2x.t[x]` and: `P ∧ Q` subtype_rel: `A ⊆r B` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` pi1: `fst(t)` isl: `isl(x)` sq_type: `SQType(T)` guard: `{T}` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cand: `A c∧ B` quotient: `x,y:A//B[x; y]` squash: `↓T`
Lemmas referenced :  nat_wf bool_wf strong-continuity2-no-inner-squash-cantor2 exists_wf int_seg_wf unit_wf2 all_wf equal_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self isect_wf assert_wf isl_wf and_wf btrue_wf subtype_base_sq bool_subtype_base nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__equal_int intformeq_wf int_formula_prop_eq_lemma le_wf true_wf quotient_wf equiv_rel_true quotient-member-eq equal-wf-base member_wf squash_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation functionEquality cut introduction extract_by_obid hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality productElimination isectElimination natural_numberEquality setElimination rename because_Cache unionEquality sqequalRule lambdaEquality productEquality applyEquality functionExtensionality independent_isectElimination independent_pairFormation inlEquality dependent_pairFormation independent_pairEquality equalityTransitivity equalitySymmetry independent_functionElimination dependent_set_memberEquality applyLambdaEquality instantiate cumulativity unionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll promote_hyp pointwiseFunctionality pertypeElimination imageElimination universeEquality imageMemberEquality baseClosed

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}M:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{}.  \mforall{}f,g:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.    ((f  =  g)  {}\mRightarrow{}  ((F  f)  =  (F  g))))

Date html generated: 2017_04_17-AM-09_57_00
Last ObjectModification: 2017_02_27-PM-05_51_30

Theory : continuity

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