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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