Nuprl Lemma : modulus-int_mod-nonzero
∀[n:ℕ+]. ∀[x:ℤ_n]. x mod n ∈ ℕ+n supposing x ≠ 0 ∈ ℤ_n
Proof
Definitions occuring in Statement :
int_mod: ℤ_n,
modulus: a mod n,
int_seg: {i..j-},
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
nequal: a ≠ b ∈ T ,
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
guard: {T},
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
false: False,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
int_mod: ℤ_n,
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_type: SQType(T)
Lemmas referenced :
modulus_wf_int_mod,
int_seg_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
nequal_wf,
int_mod_wf,
int-subtype-int_mod,
nat_plus_wf,
quotient-member-eq,
eqmod_wf,
eqmod_equiv_rel,
equal-wf-base,
equal-wf-T-base,
int_seg_wf,
subtype_base_sq,
int_subtype_base,
mod-eqmod,
eqmod_inversion
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
applyEquality,
because_Cache,
sqequalRule,
independent_pairFormation,
natural_numberEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
lambdaFormation,
pointwiseFunctionalityForEquality,
pertypeElimination,
productEquality,
baseClosed,
instantiate,
cumulativity
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbZ{}\_n]. x mod n \mmember{} \mBbbN{}\msupplus{}n supposing x \mneq{} 0 \mmember{} \mBbbZ{}\_n
Date html generated:
2018_05_21-PM-00_59_50
Last ObjectModification:
2018_05_19-AM-06_36_12
Theory : num_thy_1
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