Nuprl Lemma : append-factors
∀n,m:ℕ+. (factors(n * m) = (factors(n) + factors(m)) ∈ bag(Prime))
Proof
Definitions occuring in Statement :
factors: factors(n),
Prime: Prime,
bag-append: as + bs,
bag: bag(T),
nat_plus: ℕ+,
all: ∀x:A. B[x],
multiply: n * m,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
inject: Inj(A;B;f),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
nat_plus: ℕ+,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
Prime: Prime,
int_upper: {i...},
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
prime-product-injection,
factors_wf,
mul_nat_plus,
bag-append_wf,
Prime_wf,
bag-product-primes,
equal_wf,
squash_wf,
true_wf,
product-factors,
int-bag-product_wf,
subtype_rel_bag,
iff_weakening_equal,
int-bag-product-append,
less_than_wf,
nat_plus_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
independent_functionElimination,
dependent_set_memberEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
intEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
multiplyEquality
Latex:
\mforall{}n,m:\mBbbN{}\msupplus{}. (factors(n * m) = (factors(n) + factors(m)))
Date html generated:
2018_05_21-PM-07_23_28
Last ObjectModification:
2017_07_26-PM-05_06_22
Theory : general
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