### Nuprl Lemma : absval_zero

`∀[i:ℤ]. uiff(|i| = 0 ∈ ℤ;i = 0 ∈ ℤ)`

Proof

Definitions occuring in Statement :  absval: `|i|` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` 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` prop: `ℙ` subtype_rel: `A ⊆r B` bfalse: `ff` exists: `∃x:A. B[x]` 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 :  absval_unfold2 lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf equal-wf-base int_subtype_base eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot minus-one-mul add-commutes minus-one-mul-top add-mul-special zero-mul zero-add minus-zero
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis 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 intEquality applyEquality dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity impliesFunctionality independent_pairEquality axiomEquality baseApply closedConclusion addEquality lambdaEquality minusEquality

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

Date html generated: 2017_04_14-AM-07_17_01
Last ObjectModification: 2017_02_27-PM-02_51_43

Theory : arithmetic

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