Nuprl Lemma : mul_cancel_in_eq
∀[a,b:ℤ]. ∀[n:ℤ-o]. a = b ∈ ℤ supposing (n * a) = (n * b) ∈ ℤ
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat_plus: ℕ+,
gt: i > j,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
guard: {T},
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T
Lemmas referenced :
equal_wf,
int_nzero_wf,
nat_plus_wf,
decidable__lt,
or_wf,
less_than_wf,
false_wf,
not-gt-2,
decidable__int_equal,
not-equal-2,
not-lt-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
add-associates,
gt_wf,
mul_preserves_lt,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
minus-one-mul,
mul-associates,
minus-one-mul-top,
mul-swap,
mul-commutes,
one-mul,
subtract_wf,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
add-zero,
subtype_base_sq,
int_subtype_base,
not-equal-implies-less,
subtype_rel_self,
less-iff-le,
le_reflexive,
zero-add,
minus-zero,
omega-shadow,
int_nzero_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
isect_memberFormation,
sqequalRule,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_functionElimination,
unionElimination,
inlFormation,
independent_pairFormation,
voidElimination,
productElimination,
independent_isectElimination,
inrFormation,
addEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
voidEquality,
independent_functionElimination,
addLevel,
orFunctionality,
dependent_set_memberEquality,
minusEquality,
promote_hyp,
instantiate,
cumulativity,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. a = b supposing (n * a) = (n * b)
Date html generated:
2017_04_14-AM-07_20_22
Last ObjectModification:
2017_02_27-PM-02_53_57
Theory : arithmetic
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