Nuprl Lemma : residues-equal-bags
∀n:{2...}. ∀a:ℕ+. (CoPrime(n,a) ⇒ (map(λi.(ai mod n);residues-mod(n)) = residues-mod(n) ∈ bag(residue(n))))
Proof
Definitions occuring in Statement :
residues-mod: residues-mod(n),
residue-mul: (ai mod n),
residue: residue(n),
bag: bag(T),
coprime: CoPrime(a,b),
map: map(f;as),
int_upper: {i...},
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag: bag(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
residue: residue(n),
int_upper: {i...},
int_seg: {i..j-},
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
nat_plus: ℕ+
Lemmas referenced :
int_upper_wf,
nat_plus_wf,
coprime_wf,
residues-permute,
residues-mod_wf,
sq_stable__coprime,
residue-mul_wf,
map_wf,
permutation-equiv,
permutation_wf,
residue_wf,
list_wf,
quotient-member-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
lambdaEquality,
independent_isectElimination,
independent_pairFormation,
setElimination,
rename,
independent_functionElimination,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality
Latex:
\mforall{}n:\{2...\}. \mforall{}a:\mBbbN{}\msupplus{}. (CoPrime(n,a) {}\mRightarrow{} (map(\mlambda{}i.(ai mod n);residues-mod(n)) = residues-mod(n)))
Date html generated:
2016_05_15-PM-07_31_54
Last ObjectModification:
2016_01_16-AM-09_36_25
Theory : general
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