### Nuprl Lemma : mul_positive

`∀a,b:ℤ.  (0 < a `` 0 < b `` 0 < a * b)`

Proof

Definitions occuring in Statement :  less_than: `a < b` all: `∀x:A. B[x]` implies: `P `` Q` multiply: `n * m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` prop: `ℙ` squash: `↓T` member: `t ∈ T` less_than': `less_than'(a;b)` cand: `A c∧ B` and: `P ∧ Q` less_than: `a < b` implies: `P `` Q` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` top: `Top` true: `True` 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`
Lemmas referenced :  less_than_wf 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 bool_wf iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot istype-assert
Rules used in proof :  intEquality isectElimination extract_by_obid sqequalHypSubstitution baseClosed thin imageMemberEquality sqequalRule introduction hypothesisEquality multiplyEquality natural_numberEquality hypothesis independent_pairFormation cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution multiplyPositive Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination equalityElimination because_Cache productElimination independent_isectElimination lessCases Error :isect_memberFormation_alt,  axiomSqEquality Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  voidElimination imageElimination independent_functionElimination Error :dependent_pairFormation_alt,  equalityTransitivity equalitySymmetry Error :equalityIsType4,  baseApply closedConclusion applyEquality promote_hyp dependent_functionElimination instantiate cumulativity Error :functionIsType,  Error :equalityIsType1

Latex:
\mforall{}a,b:\mBbbZ{}.    (0  <  a  {}\mRightarrow{}  0  <  b  {}\mRightarrow{}  0  <  a  *  b)

Date html generated: 2019_06_20-AM-11_23_15
Last ObjectModification: 2018_10_16-PM-00_58_40

Theory : arithmetic

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