### Nuprl Lemma : rem_invariant

`∀[a,b:ℕ]. ∀[n:ℕ+].  ((a + (b * n) rem n) = (a rem n) ∈ ℤ)`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` nat: `ℕ` uall: `∀[x:A]. B[x]` remainder: `n rem m` multiply: `n * m` add: `n + m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` nat_plus: `ℕ+` nequal: `a ≠ b ∈ T ` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` guard: `{T}` uiff: `uiff(P;Q)` subtract: `n - m` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` true: `True` squash: `↓T`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf zero-mul add-zero nat_plus_properties intformeq_wf int_formula_prop_eq_lemma equal-wf-base int_subtype_base decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_plus_wf nat_wf lt_to_le_rw mul_preserves_le nat_plus_subtype_nat int_term_value_mul_lemma int_term_value_add_lemma itermMultiply_wf itermAdd_wf satisfiable-full-omega-tt add_functionality_wrt_le add-associates minus-one-mul add-commutes mul-distributes-right iff_weakening_equal le_wf rem_rec_case true_wf squash_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation axiomEquality remainderEquality because_Cache applyEquality baseClosed unionElimination productElimination addEquality computeAll multiplyEquality minusEquality imageMemberEquality dependent_set_memberEquality universeEquality equalitySymmetry equalityTransitivity imageElimination

Latex:
\mforall{}[a,b:\mBbbN{}].  \mforall{}[n:\mBbbN{}\msupplus{}].    ((a  +  (b  *  n)  rem  n)  =  (a  rem  n))

Date html generated: 2019_06_20-PM-01_14_58
Last ObjectModification: 2018_09_17-PM-05_47_10

Theory : int_2

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