### Nuprl Lemma : absval_ubound

`∀[i:ℤ]. ∀[n:ℕ].  uiff(|i| ≤ n;((-n) ≤ i) ∧ (i ≤ n))`

Proof

Definitions occuring in Statement :  absval: `|i|` nat: `ℕ` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` and: `P ∧ Q` minus: `-n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` ge: `i ≥ j ` uimplies: `b supposing a` all: `∀x:A. B[x]` uiff: `uiff(P;Q)` and: `P ∧ Q` le: `A ≤ B` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` squash: `↓T` not: `¬A` false: `False` cand: `A c∧ B` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` prop: `ℙ` rev_uimplies: `rev_uimplies(P;Q)` subtype_rel: `A ⊆r B` top: `Top`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename natural_numberEquality minusEquality because_Cache independent_isectElimination dependent_functionElimination productElimination independent_pairEquality isect_memberEquality_alt equalityTransitivity equalitySymmetry isectIsTypeImplies inhabitedIsType multiplyEquality Error :memTop,  lambdaFormation_alt unionElimination equalityElimination lessCases axiomSqEquality independent_pairFormation imageMemberEquality baseClosed imageElimination voidElimination independent_functionElimination addEquality productIsType dependent_pairFormation_alt equalityIstype promote_hyp instantiate cumulativity functionIsType universeIsType baseApply closedConclusion applyEquality intEquality voidEquality isect_memberEquality lambdaEquality

Latex:
\mforall{}[i:\mBbbZ{}].  \mforall{}[n:\mBbbN{}].    uiff(|i|  \mleq{}  n;((-n)  \mleq{}  i)  \mwedge{}  (i  \mleq{}  n))

Date html generated: 2020_05_19-PM-09_35_23
Last ObjectModification: 2020_01_04-PM-07_56_38

Theory : arithmetic

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