### Nuprl Lemma : absval_mul

`∀[x,y:ℤ].  (|x * y| = (|x| * |y|) ∈ ℤ)`

Proof

Definitions occuring in Statement :  absval: `|i|` uall: `∀[x:A]. B[x]` multiply: `n * m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` false: `False` bfalse: `ff` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` prop: `ℙ` le: `A ≤ B` nat: `ℕ` subtract: `n - m` nat_plus: `ℕ+` decidable: `Dec(P)` cand: `A c∧ B`
Lemmas referenced :  absval_unfold2 lt_int_wf eqtt_to_assert assert_of_lt_int istype-top istype-void eqff_to_assert int_subtype_base bool_subtype_base bool_cases_sqequal subtype_base_sq iff_transitivity assert_wf bnot_wf not_wf less_than_wf iff_weakening_uiff assert_of_bnot istype-less_than istype-assert not-lt-2 minus-one-mul mul-associates istype-int minus-one-mul-top mul-swap one-mul bool_wf less_than_irreflexivity less_than_transitivity1 le_wf le_weakening2 mul_preserves_le mul-commutes zero-mul le-add-cancel zero-add add-associates add_functionality_wrt_le add-commutes add-zero minus-zero minus-add condition-implies-le less-iff-le mul_preserves_lt decidable__int_equal decidable__lt istype-false not-equal-2 add_functionality_wrt_lt le_reflexive add-mul-special
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin multiplyEquality hypothesisEquality hypothesis natural_numberEquality Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination because_Cache lessCases axiomSqEquality Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  independent_pairFormation voidElimination imageMemberEquality baseClosed imageElimination independent_functionElimination Error :dependent_pairFormation_alt,  Error :equalityIsType4,  baseApply closedConclusion applyEquality promote_hyp dependent_functionElimination instantiate Error :functionIsType,  Error :universeIsType,  Error :equalityIsType1,  minusEquality Error :lambdaEquality_alt,  cumulativity axiomEquality voidEquality isect_memberEquality dependent_set_memberEquality intEquality lambdaEquality addEquality

Latex:
\mforall{}[x,y:\mBbbZ{}].    (|x  *  y|  =  (|x|  *  |y|))

Date html generated: 2019_06_20-AM-11_24_38
Last ObjectModification: 2018_10_27-AM-11_38_11

Theory : arithmetic

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