### Nuprl Lemma : bag-member-two-factorizations

`∀[n:ℕ]. ∀[a,b:ℤ].  uiff(<a, b> ↓∈ two-factorizations(n);(1 ≤ a) ∧ (a ≤ n) ∧ ((a * b) = n ∈ ℤ))`

Proof

Definitions occuring in Statement :  two-factorizations: `two-factorizations(n)` bag-member: `x ↓∈ bs` nat: `ℕ` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` and: `P ∧ Q` pair: `<a, b>` product: `x:A × B[x]` multiply: `n * m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` pi2: `snd(t)` pi1: `fst(t)` nat: `ℕ` prop: `ℙ` iff: `P `⇐⇒` Q` le: `A ≤ B` not: `¬A` false: `False` rev_implies: `P `` Q` bag-member: `x ↓∈ bs` squash: `↓T` two-factorizations: `two-factorizations(n)` nequal: `a ≠ b ∈ T ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` guard: `{T}` less_than: `a < b` cand: `A c∧ B` int_nzero: `ℤ-o` sq_stable: `SqStable(P)` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` rev_uimplies: `rev_uimplies(P;Q)` divides: `b | a` nat_plus: `ℕ+` less_than': `less_than'(a;b)` true: `True` gt: `i > j` div_nrel: `Div(a;n;q)` lelt: `i ≤ j < k`
Lemmas referenced :  bag-member-list decidable__equal_product decidable__equal_int two-factorizations_wf subtype_rel_list equal_wf less_than'_wf bag-member_wf list-subtype-bag le_wf equal-wf-base-T uiff_wf l_member_wf int_subtype_base nat_wf member-mapfilter less_than_wf from-upto_wf set_wf eq_int_wf nat_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf equal-wf-base equal-wf-T-base assert_wf mapfilter_wf int_nzero_wf subtype_rel_sets nequal_wf int_nzero_properties intformnot_wf int_formula_prop_not_lemma exists_wf sq_stable__le decidable__le intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma assert_of_eq_int subtype_base_sq div_rem_sum itermMultiply_wf int_term_value_mul_lemma decidable__lt member-from-upto divides_iff_rem_zero div_unique2 false_wf not-lt-2 add_functionality_wrt_le add-commutes zero-add le-add-cancel pos_mul_arg_bounds intformimplies_wf intformor_wf int_formual_prop_imp_lemma int_formula_prop_or_lemma
Rules used in proof :  cut addLevel sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity productElimination thin independent_pairFormation isect_memberFormation introduction independent_isectElimination extract_by_obid isectElimination productEquality intEquality independent_functionElimination lambdaFormation because_Cache sqequalRule lambdaEquality dependent_functionElimination hypothesisEquality hypothesis independent_pairEquality applyEquality setEquality multiplyEquality setElimination rename voidElimination cumulativity natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry universeEquality imageElimination imageMemberEquality baseClosed instantiate baseApply closedConclusion isect_memberEquality addEquality dependent_set_memberEquality remainderEquality dependent_pairFormation int_eqEquality voidEquality computeAll divideEquality applyLambdaEquality unionElimination

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[a,b:\mBbbZ{}].    uiff(<a,  b>  \mdownarrow{}\mmember{}  two-factorizations(n);(1  \mleq{}  a)  \mwedge{}  (a  \mleq{}  n)  \mwedge{}  ((a  *  b)  =  n))

Date html generated: 2018_05_21-PM-09_06_14
Last ObjectModification: 2017_07_26-PM-06_28_59

Theory : general

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