### Nuprl Lemma : absval_le_zero

`∀[i:ℤ]. uiff(|i| ≤ 0;i = 0 ∈ ℤ)`

Proof

Definitions occuring in Statement :  absval: `|i|` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` natural_number: `\$n` 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` le: `A ≤ B` cand: `A c∧ B` guard: `{T}` subtype_rel: `A ⊆r B` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` prop: `ℙ` decidable: `Dec(P)` subtract: `n - m`
Lemmas referenced :  absval_unfold2 lt_int_wf eqtt_to_assert assert_of_lt_int istype-top istype-void less_than_transitivity1 less_than_irreflexivity istype-le le_weakening le_witness_for_triv int_subtype_base eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf less_than_wf iff_weakening_uiff assert_of_bnot istype-less_than istype-assert istype-int add_functionality_wrt_le le_reflexive decidable__int_equal istype-false not-equal-2 condition-implies-le minus-zero add-zero add-associates minus-add minus-minus minus-one-mul zero-add minus-one-mul-top two-mul add-commutes mul-distributes-right one-mul le-add-cancel not-lt-2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin 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 dependent_functionElimination Error :equalityIstype,  applyEquality sqequalBase Error :dependent_pairFormation_alt,  promote_hyp instantiate cumulativity Error :functionIsType,  Error :universeIsType,  independent_pairEquality axiomEquality minusEquality addEquality multiplyEquality intEquality

Latex:
\mforall{}[i:\mBbbZ{}].  uiff(|i|  \mleq{}  0;i  =  0)

Date html generated: 2019_06_20-AM-11_24_30
Last ObjectModification: 2019_02_12-PM-01_59_51

Theory : arithmetic

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