Nuprl Lemma : p-adic-inv-lemma
∀p:{p:{2...}| prime(p)} . ∀a:{a:p-adics(p)| ¬((a 1) = 0 ∈ ℤ)} . ∀n:ℕ+. (∃c:ℕp^n [((c * (a n)) ≡ 1 mod p^n)])
Proof
Definitions occuring in Statement :
p-adics: p-adics(p),
eqmod: a ≡ b mod m,
prime: prime(a),
exp: i^n,
int_upper: {i...},
int_seg: {i..j-},
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
subtract: n - m,
spreadn: spread7,
genrec-ap: genrec-ap,
p-reduce: i mod(p^n),
modulus: a mod n,
p-adic-inv-lemma1,
gcd-reduce-coprime,
gcd-reduce-ext,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a
Lemmas referenced :
p-adic-inv-lemma1,
lifting-strict-spread,
istype-void,
strict4-spread,
gcd-reduce-coprime,
gcd-reduce-ext
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}p:\{p:\{2...\}| prime(p)\} . \mforall{}a:\{a:p-adics(p)| \mneg{}((a 1) = 0)\} . \mforall{}n:\mBbbN{}\msupplus{}.
(\mexists{}c:\mBbbN{}p\^{}n [((c * (a n)) \mequiv{} 1 mod p\^{}n)])
Date html generated:
2019_10_15-AM-10_34_47
Last ObjectModification:
2019_06_20-PM-06_50_52
Theory : rings_1
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