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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