Nuprl Lemma : div_minus
∀[a:ℤ]. ∀[b:ℤ-o]. (((-a) ÷ -b) = (a ÷ b) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
minus: -n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
int_nzero: ℤ-o,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
or: P ∨ Q,
guard: {T},
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
false: False,
prop: ℙ,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not-equal-2,
le_antisymmetry_iff,
condition-implies-le,
minus-zero,
add-zero,
add-associates,
minus-add,
minus-minus,
minus-one-mul,
zero-add,
minus-one-mul-top,
two-mul,
add-commutes,
mul-distributes-right,
one-mul,
add_functionality_wrt_le,
le-add-cancel,
add-swap,
add-mul-special,
equal-wf-T-base,
int_nzero_wf,
equal_wf,
squash_wf,
true_wf,
div_anti_sym,
subtype_rel_self,
iff_weakening_equal,
div_anti_sym2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
addEquality,
hypothesis,
because_Cache,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
unionElimination,
isectElimination,
minusEquality,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
multiplyEquality,
independent_functionElimination,
baseClosed,
axiomEquality,
intEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
divideEquality,
imageMemberEquality,
instantiate
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}]. (((-a) \mdiv{} -b) = (a \mdiv{} b))
Date html generated:
2019_06_20-AM-11_25_12
Last ObjectModification:
2018_08_18-PM-00_47_28
Theory : arithmetic
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