Nuprl Lemma : int-eq-constraint-factor
∀[a:ℤ]. ∀[g:ℤ-o]. ∀[xs,L:ℤ List].
uiff([1 / xs] ⋅ [a / g * L] = 0 ∈ ℤ;((a rem g) = 0 ∈ ℤ) ∧ ([1 / xs] ⋅ [a ÷ g / L] = 0 ∈ ℤ))
Proof
Definitions occuring in Statement :
int-vec-mul: a * as,
integer-dot-product: as ⋅ bs,
cons: [a / b],
list: T List,
int_nzero: ℤ-o,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
remainder: n rem m,
divide: n ÷ m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
false: False,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
int_dot_cons_lemma,
equal-wf-T-base,
integer-dot-product_wf,
cons_wf,
int-vec-mul_wf,
equal_wf,
list_wf,
int_nzero_wf,
int-dot-mul-right,
subtype_base_sq,
int_subtype_base,
add-associates,
minus-one-mul,
one-mul,
zero-add,
add-commutes,
add-mul-special,
zero-mul,
add-zero,
minus-one-mul-top,
mul-swap,
squash_wf,
true_wf,
rem-exact,
iff_weakening_equal,
mul_cancel_in_eq,
mul-distributes,
mul-commutes,
div_rem_sum,
mul_preserves_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
productElimination,
independent_pairEquality,
axiomEquality,
isectElimination,
intEquality,
natural_numberEquality,
hypothesisEquality,
setElimination,
rename,
baseClosed,
productEquality,
because_Cache,
remainderEquality,
lambdaFormation,
independent_functionElimination,
divideEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
minusEquality,
multiplyEquality,
instantiate,
cumulativity,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
hyp_replacement
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[g:\mBbbZ{}\msupminus{}\msupzero{}]. \mforall{}[xs,L:\mBbbZ{} List].
uiff([1 / xs] \mcdot{} [a / g * L] = 0;((a rem g) = 0) \mwedge{} ([1 / xs] \mcdot{} [a \mdiv{} g / L] = 0))
Date html generated:
2017_04_14-AM-08_56_12
Last ObjectModification:
2017_02_27-PM-03_39_50
Theory : omega
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