### Nuprl Lemma : absval-positive

`∀[x:ℤ]. uiff(0 < |x|;x ≠ 0)`

Proof

Definitions occuring in Statement :  absval: `|i|` less_than: `a < b` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` nequal: `a ≠ b ∈ T ` natural_number: `\$n` int: `ℤ`
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` prop: `ℙ` nequal: `a ≠ b ∈ T ` guard: `{T}` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` le: `A ≤ B` subtract: `n - m` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` rev_uimplies: `rev_uimplies(P;Q)` nat: `ℕ` gt: `i > j`
Lemmas referenced :  absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf less_than_transitivity1 le_weakening less_than_irreflexivity equal_wf equal-wf-base int_subtype_base decidable__lt false_wf not-lt-2 not-equal-2 less-iff-le add_functionality_wrt_le add-associates add-swap add-commutes zero-add le-add-cancel condition-implies-le minus-add minus-zero add-zero nequal_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot decidable__le le_wf not-le-2 or_wf absval_wf nat_wf member-less_than not-gt-2 subtract_wf le_reflexive minus-one-mul-top minus-le le-add-cancel-alt minus-one-mul add-mul-special zero-mul
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis minusEquality natural_numberEquality lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination because_Cache lessCases sqequalAxiom isect_memberEquality independent_pairFormation voidElimination voidEquality imageMemberEquality baseClosed imageElimination independent_functionElimination dependent_functionElimination intEquality applyEquality lambdaEquality addEquality dependent_pairFormation promote_hyp instantiate cumulativity impliesFunctionality inlFormation inrFormation addLevel orFunctionality independent_pairEquality setElimination rename multiplyEquality

Latex:
\mforall{}[x:\mBbbZ{}].  uiff(0  <  |x|;x  \mneq{}  0)

Date html generated: 2017_04_14-AM-07_16_57
Last ObjectModification: 2017_02_27-PM-02_51_48

Theory : arithmetic

Home Index