Nuprl Lemma : zero-div-rem
∀[x:ℤ-o]. ((0 ÷ x ~ 0) ∧ (0 rem x ~ 0))
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
remainder: n rem m,
divide: n ÷ m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
int_nzero: ℤ-o,
prop: ℙ,
nat: ℕ,
uiff: uiff(P;Q),
true: True,
squash: ↓T,
not: ¬A,
false: False,
absval: |i|,
less_than': less_than'(a;b),
le: A ≤ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
nequal: a ≠ b ∈ T
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
rem-zero,
iff_weakening_equal,
int_nzero_wf,
add-zero,
zero-mul,
le_wf,
less_than_wf,
absval_wf,
nat_wf,
equal-wf-base-T,
div_unique3,
absval_pos,
true_wf,
squash_wf,
equal_wf,
false_wf,
set_subtype_base,
absval-positive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
sqequalIntensionalEquality,
hypothesisEquality,
applyEquality,
sqequalRule,
baseClosed,
productElimination,
independent_pairEquality,
axiomSqEquality,
natural_numberEquality,
dependent_pairFormation,
because_Cache,
multiplyEquality,
setElimination,
rename,
lambdaFormation,
productEquality,
lambdaEquality,
addEquality,
functionEquality,
imageMemberEquality,
universeEquality,
imageElimination,
dependent_set_memberEquality,
voidElimination
Latex:
\mforall{}[x:\mBbbZ{}\msupminus{}\msupzero{}]. ((0 \mdiv{} x \msim{} 0) \mwedge{} (0 rem x \msim{} 0))
Date html generated:
2019_06_20-AM-11_24_56
Last ObjectModification:
2018_08_21-PM-10_43_08
Theory : arithmetic
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