Nuprl Lemma : positive-prime-divides-product
∀p:{p:ℕ| prime(p)} . ∀qs:{p:ℕ| prime(p)} List. ((p | reduce(λx,y. (x * y);1;qs)) ⇒ (p ∈ qs))
Proof
Definitions occuring in Statement :
prime: prime(a),
divides: b | a,
l_member: (x ∈ l),
reduce: reduce(f;k;as),
list: T List,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
top: Top,
prime: prime(a),
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
not: ¬A,
assoced: a ~ b,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
positive-prime-divides-prime,
equal_wf,
cons_member,
one_divs_any,
decidable__equal_nat,
decidable__equal_set,
sq_stable__l_member,
cons_wf,
nil_wf,
set_wf,
reduce_cons_lemma,
reduce_nil_lemma,
list_wf,
l_member_wf,
reduce_wf,
divides_wf,
prime_wf,
nat_wf,
list_induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
setEquality,
hypothesis,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
intEquality,
multiplyEquality,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
dependent_set_memberEquality,
addLevel,
productElimination,
levelHypothesis,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
unionElimination,
inlFormation,
inrFormation,
independent_isectElimination
Latex:
\mforall{}p:\{p:\mBbbN{}| prime(p)\} . \mforall{}qs:\{p:\mBbbN{}| prime(p)\} List. ((p | reduce(\mlambda{}x,y. (x * y);1;qs)) {}\mRightarrow{} (p \mmember{} qs))
Date html generated:
2016_05_14-PM-04_27_17
Last ObjectModification:
2016_01_14-PM-11_37_40
Theory : num_thy_1
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