Nuprl Lemma : p-mul-int
∀[p:ℕ+]. ∀[k,j:ℤ]. (k(p) * j(p) = k * j(p) ∈ p-adics(p))
Proof
Definitions occuring in Statement :
p-int: k(p),
p-mul: x * y,
p-adics: p-adics(p),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
p-int: k(p),
p-mul: x * y,
p-reduce: i mod(p^n),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_seg: {i..j-},
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
p-adics-equal,
p-mul_wf,
p-int_wf,
nat_plus_wf,
exp_wf2,
nat_plus_subtype_nat,
modulus_wf_int_mod,
exp_wf_nat_plus,
int-subtype-int_mod,
int_seg_wf,
eqmod_functionality_wrt_eqmod,
eqmod_transitivity,
mod-eqmod,
multiply_functionality_wrt_eqmod,
eqmod_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
multiplyEquality,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
intEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
applyEquality,
setElimination,
rename,
lambdaEquality,
natural_numberEquality,
independent_functionElimination
Latex:
\mforall{}[p:\mBbbN{}\msupplus{}]. \mforall{}[k,j:\mBbbZ{}]. (k(p) * j(p) = k * j(p))
Date html generated:
2018_05_21-PM-03_18_57
Last ObjectModification:
2018_05_19-AM-08_09_57
Theory : rings_1
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