### Nuprl Lemma : div_floor_mod_sum

`∀[a:ℤ]. ∀[n:ℕ+].  (a = (((a ÷↓ n) * n) + (a mod n)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  div_floor: `a ÷↓ n` modulus: `a mod n` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` multiply: `n * m` add: `n + m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` modulus: `a mod n` div_floor: `a ÷↓ n` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` guard: `{T}` sq_type: `SQType(T)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` nat: `ℕ` has-value: `(a)↓` subtract: `n - m`
Lemmas referenced :  div_rem_sum subtype_rel_sets less_than_wf nequal_wf less_than_transitivity1 le_weakening less_than_irreflexivity equal_wf equal-wf-base int_subtype_base rem_bounds_z subtype_base_sq lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_transitivity2 le_weakening2 eqff_to_assert bool_cases_sqequal bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot equal-wf-base-T absval_wf nat_wf nat_plus_wf value-type-has-value int-value-type subtract_wf nat_plus_subtype_nat mul-commutes add-commutes mul-distributes-right add-associates minus-one-mul-top add-swap add-mul-special zero-mul zero-add squash_wf true_wf add_functionality_wrt_eq absval_pos iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality sqequalRule intEquality because_Cache lambdaEquality natural_numberEquality hypothesis independent_isectElimination setElimination rename setEquality lambdaFormation dependent_functionElimination independent_functionElimination voidElimination baseClosed remainderEquality divideEquality instantiate cumulativity equalityTransitivity equalitySymmetry unionElimination equalityElimination productElimination lessCases sqequalAxiom isect_memberEquality independent_pairFormation voidEquality imageMemberEquality imageElimination dependent_pairFormation promote_hyp impliesFunctionality addEquality multiplyEquality axiomEquality callbyvalueReduce equalityUniverse levelHypothesis minusEquality universeEquality

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[n:\mBbbN{}\msupplus{}].    (a  =  (((a  \mdiv{}\mdownarrow{}  n)  *  n)  +  (a  mod  n)))

Date html generated: 2017_04_14-AM-07_19_33
Last ObjectModification: 2017_02_27-PM-02_53_18

Theory : arithmetic

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