Nuprl Lemma : div_anti_sym
∀[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,
decidable: Dec(P),
nat: ℕ,
int_lower: {...i},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
nat_plus: ℕ+
Lemmas referenced :
int_nzero_wf,
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,
decidable__le,
div_4_to_1,
le_wf,
false_wf,
not-le-2,
le-add-cancel2,
subtract_wf,
le_reflexive,
mul-associates,
zero-mul,
subtype_base_sq,
int_subtype_base,
div_3_to_1,
div_2_to_1,
decidable__lt,
not-lt-2,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
lemma_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
intEquality,
lambdaFormation,
addEquality,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
productElimination,
independent_isectElimination,
unionElimination,
minusEquality,
applyEquality,
lambdaEquality,
voidElimination,
voidEquality,
multiplyEquality,
independent_functionElimination,
dependent_set_memberEquality,
divideEquality,
independent_pairFormation,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}]. ((a \mdiv{} -b) = (-(a \mdiv{} b)))
Date html generated:
2016_05_13-PM-03_36_14
Last ObjectModification:
2015_12_26-AM-09_44_01
Theory : arithmetic
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