Nuprl Lemma : rem_eq_args_z

`∀[a:ℤ]. ∀[b:ℤ-o].  (a rem b) = 0 ∈ ℤ supposing |a| = |b| ∈ ℤ`

Proof

Definitions occuring in Statement :  absval: `|i|` int_nzero: `ℤ-o` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` remainder: `n rem m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` uimplies: `b supposing a` int_nzero: `ℤ-o` nat: `ℕ` 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` not: `¬A` false: `False` nequal: `a ≠ b ∈ T ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` decidable: `Dec(P)` int_lower: `{...i}`
Lemmas referenced :  equal-wf-base-T int_subtype_base nat_plus_wf absval_wf nat_wf int_nzero_wf absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf equal_wf squash_wf true_wf nat_plus_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf equal-wf-base iff_weakening_equal rem_eq_args less_than_transitivity1 le_weakening eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot minus_mono_wrt_eq itermMinus_wf int_term_value_minus_lemma rem_antisym nequal_wf minus-zero decidable__lt absval_pos int_nzero_properties decidable__le intformnot_wf intformle_wf int_formula_prop_not_lemma int_formula_prop_le_lemma le_wf rem_sym absval_neg
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis introduction extract_by_obid sqequalHypSubstitution isectElimination thin intEquality sqequalRule baseApply closedConclusion baseClosed hypothesisEquality applyEquality setElimination rename isect_memberFormation because_Cache lambdaEquality isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry minusEquality natural_numberEquality unionElimination equalityElimination productElimination independent_isectElimination lessCases sqequalAxiom independent_pairFormation voidElimination voidEquality imageMemberEquality imageElimination independent_functionElimination universeEquality equalityUniverse levelHypothesis remainderEquality dependent_pairFormation int_eqEquality dependent_functionElimination computeAll dependent_set_memberEquality promote_hyp instantiate cumulativity

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}].    (a  rem  b)  =  0  supposing  |a|  =  |b|

Date html generated: 2017_04_14-AM-09_16_42
Last ObjectModification: 2017_02_27-PM-03_53_50

Theory : int_2

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