### Nuprl Lemma : rem-zero-implies-minus

`∀x:ℤ. ∀y:ℤ-o.  (((x rem y) = 0 ∈ ℤ) `` ((-x rem y) = 0 ∈ ℤ))`

Proof

Definitions occuring in Statement :  int_nzero: `ℤ-o` all: `∀x:A. B[x]` implies: `P `` Q` remainder: `n rem m` minus: `-n` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` decidable: `Dec(P)` or: `P ∨ Q` false: `False` uiff: `uiff(P;Q)` and: `P ∧ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` prop: `ℙ` sq_type: `SQType(T)` guard: `{T}` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  div_rem_sum subtype_base_sq int_subtype_base int_nzero_properties decidable__equal_int add-is-int-iff multiply-is-int-iff full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermVar_wf itermMultiply_wf itermAdd_wf itermConstant_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_mul_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_wf false_wf minus-one-mul divide_wfa mul-commutes mul-swap rem-exact set_subtype_base nequal_wf int_nzero_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality instantiate cumulativity intEquality independent_isectElimination hypothesis setElimination rename dependent_functionElimination because_Cache unionElimination equalityTransitivity equalitySymmetry pointwiseFunctionality promote_hyp sqequalRule baseApply closedConclusion baseClosed productElimination natural_numberEquality approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :universeIsType,  multiplyEquality minusEquality Error :equalityIstype,  Error :inhabitedIsType,  applyEquality sqequalBase

Latex:
\mforall{}x:\mBbbZ{}.  \mforall{}y:\mBbbZ{}\msupminus{}\msupzero{}.    (((x  rem  y)  =  0)  {}\mRightarrow{}  ((-x  rem  y)  =  0))

Date html generated: 2019_06_20-PM-02_24_48
Last ObjectModification: 2019_03_06-AM-11_06_26

Theory : num_thy_1

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