Nuprl Lemma : mul_cancel_in_lt
∀[a,b:ℤ]. ∀[n:ℕ+]. a < b supposing n * a < n * b
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
gt: i > j,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
guard: {T},
implies: P ⇒ Q
Lemmas referenced :
less_than_irreflexivity,
less_than_transitivity1,
int_subtype_base,
set_subtype_base,
multiply-is-int-iff,
not-gt-2,
nat_plus_subtype_nat,
mul_preserves_le,
decidable__lt,
nat_plus_wf,
member-less_than,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
independent_isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
intEquality,
dependent_functionElimination,
unionElimination,
applyEquality,
productElimination,
baseApply,
closedConclusion,
baseClosed,
lambdaEquality,
natural_numberEquality,
independent_functionElimination,
voidElimination
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. a < b supposing n * a < n * b
Date html generated:
2016_05_13-PM-03_40_48
Last ObjectModification:
2016_01_14-PM-06_38_35
Theory : arithmetic
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