Nuprl Lemma : modulus_wf_int_mod
∀[n:ℕ+]. ∀[x:ℤ_n].  (x mod n ∈ ℕn)
Proof
Definitions occuring in Statement : 
int_mod: ℤ_n
, 
modulus: a mod n
, 
int_seg: {i..j-}
, 
nat_plus: ℕ+
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
natural_number: $n
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
int_mod: ℤ_n
, 
nat_plus: ℕ+
, 
quotient: x,y:A//B[x; y]
, 
and: P ∧ Q
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
lelt: i ≤ j < k
, 
int_seg: {i..j-}
, 
squash: ↓T
, 
prop: ℙ
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
true: True
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
int_seg_wf, 
mod_bounds, 
equal_wf, 
squash_wf, 
true_wf, 
istype-universe, 
modulus_functionality_wrt_eqmod, 
modulus_wf, 
nat_plus_inc_int_nzero, 
subtype_rel_self, 
iff_weakening_equal, 
istype-le, 
istype-less_than, 
eqmod_wf, 
istype-int, 
int_mod_wf, 
nat_plus_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
sqequalHypSubstitution, 
pointwiseFunctionalityForEquality, 
extract_by_obid, 
isectElimination, 
thin, 
natural_numberEquality, 
setElimination, 
rename, 
because_Cache, 
hypothesis, 
sqequalRule, 
pertypeElimination, 
promote_hyp, 
productElimination, 
equalityTransitivity, 
equalitySymmetry, 
inhabitedIsType, 
lambdaFormation_alt, 
independent_pairFormation, 
hypothesisEquality, 
dependent_set_memberEquality_alt, 
applyEquality, 
lambdaEquality_alt, 
imageElimination, 
universeIsType, 
instantiate, 
universeEquality, 
intEquality, 
independent_isectElimination, 
imageMemberEquality, 
baseClosed, 
independent_functionElimination, 
productIsType, 
equalityIstype, 
dependent_functionElimination, 
sqequalBase, 
axiomEquality, 
isect_memberEquality_alt, 
isectIsTypeImplies
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}].  \mforall{}[x:\mBbbZ{}\_n].    (x  mod  n  \mmember{}  \mBbbN{}n)
Date html generated:
2020_05_19-PM-10_02_24
Last ObjectModification:
2020_01_04-PM-08_03_28
Theory : num_thy_1
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