### Nuprl Lemma : mul_cancel_in_le

`∀[a,b:ℤ]. ∀[n:ℕ+].  a ≤ b supposing (n * a) ≤ (n * b)`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` multiply: `n * m` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` guard: `{T}` subtract: `n - m` top: `Top` less_than': `less_than'(a;b)` true: `True` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T `
Lemmas referenced :  less_than'_wf le_wf nat_plus_wf multiply-is-int-iff set_subtype_base less_than_wf int_subtype_base decidable__lt equal_wf false_wf not-lt-2 decidable__int_equal not-equal-2 condition-implies-le minus-add minus-one-mul add-swap minus-one-mul-top add-commutes mul-commutes add_functionality_wrt_le le-add-cancel add-associates or_wf mul_cancel_in_lt le_weakening2 mul_cancel_in_eq subtype_rel_sets nequal_wf less_than_transitivity1 le_weakening less_than_irreflexivity equal-wf-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality lambdaEquality dependent_functionElimination hypothesisEquality because_Cache extract_by_obid isectElimination hypothesis axiomEquality multiplyEquality setElimination rename isect_memberEquality equalityTransitivity equalitySymmetry intEquality voidElimination baseApply closedConclusion baseClosed applyEquality natural_numberEquality independent_isectElimination unionElimination inlFormation independent_pairFormation lambdaFormation inrFormation addEquality voidEquality minusEquality independent_functionElimination addLevel orFunctionality setEquality

Latex:
\mforall{}[a,b:\mBbbZ{}].  \mforall{}[n:\mBbbN{}\msupplus{}].    a  \mleq{}  b  supposing  (n  *  a)  \mleq{}  (n  *  b)

Date html generated: 2017_04_14-AM-07_20_28
Last ObjectModification: 2017_02_27-PM-02_53_53

Theory : arithmetic

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