Nuprl Lemma : mul_preserves_lt
∀[a,b:ℤ]. ∀[n:ℕ+]. n * a < n * b supposing a < 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,
uiff: uiff(P;Q),
and: P ∧ Q,
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
true: True
Lemmas referenced :
less-iff-positive,
less_than_wf,
member-less_than,
nat_plus_wf,
mul_positive,
subtract_wf,
squash_wf,
true_wf,
minus-one-mul,
mul-commutes,
add-commutes,
minus-one-mul-top,
mul-distributes,
mul-swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
multiplyEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
voidElimination,
voidEquality,
minusEquality,
addEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. n * a < n * b supposing a < b
Date html generated:
2019_06_20-AM-11_23_22
Last ObjectModification:
2019_01_27-PM-03_10_12
Theory : arithmetic
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