`∀[i,j:ℕ]. ∀[n:ℕ+].  (((i rem n) + (j rem n) rem n) = (i + j rem n) ∈ ℤ)`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` nat: `ℕ` uall: `∀[x:A]. B[x]` remainder: `n rem m` add: `n + m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` true: `True` so_apply: `x[s]` so_lambda: `λ2x.t[x]` int_nzero: `ℤ-o` subtype_rel: `A ⊆r B` prop: `ℙ` and: `P ∧ Q` top: `Top` all: `∀x:A. B[x]` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` uimplies: `b supposing a` implies: `P `` Q` not: `¬A` ge: `i ≥ j ` nequal: `a ≠ b ∈ T ` nat_plus: `ℕ+` nat: `ℕ` squash: `↓T` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)`
Lemmas referenced :  nat_plus_wf nat_wf nequal_wf less_than_wf subtype_rel_sets int_subtype_base equal-wf-base int_formula_prop_wf int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_and_lemma intformless_wf itermConstant_wf itermVar_wf intformeq_wf intformand_wf full-omega-unsat nat_properties nat_plus_properties equal_wf squash_wf true_wf add_functionality_wrt_eq div_rem_sum subtype_rel_self iff_weakening_equal add-commutes add-swap mul-commutes mul-distributes-right add-associates add_nat_wf remainder_wf decidable__le add-is-int-iff intformnot_wf intformle_wf itermAdd_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_add_lemma false_wf le_wf divide_wf rem_invariant
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut hypothesis Error :universeIsType,  extract_by_obid sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality axiomEquality because_Cache Error :inhabitedIsType,  setEquality divideEquality multiplyEquality baseClosed applyEquality independent_pairFormation voidEquality voidElimination dependent_functionElimination int_eqEquality lambdaEquality dependent_pairFormation independent_functionElimination approximateComputation independent_isectElimination natural_numberEquality lambdaFormation rename setElimination addEquality remainderEquality intEquality imageElimination equalityTransitivity equalitySymmetry universeEquality imageMemberEquality instantiate productElimination dependent_set_memberEquality applyLambdaEquality unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion

Latex:
\mforall{}[i,j:\mBbbN{}].  \mforall{}[n:\mBbbN{}\msupplus{}].    (((i  rem  n)  +  (j  rem  n)  rem  n)  =  (i  +  j  rem  n))

Date html generated: 2019_06_20-PM-01_15_02
Last ObjectModification: 2018_09_26-PM-02_36_46

Theory : int_2

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