Nuprl Lemma : equal_int_mod_iff_modulus
∀[n:ℕ+]. ∀[x,y:ℤ_n]. uiff((x mod n) = (y mod n) ∈ ℤ;x = y ∈ ℤ_n)
Proof
Definitions occuring in Statement :
int_mod: ℤ_n,
modulus: a mod n,
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat_plus: ℕ+,
int_mod: ℤ_n,
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
modulus_wf_int_mod,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
int_mod_wf,
nat_plus_wf,
int-subtype-int_mod,
equal-wf-base-T,
int_subtype_base,
set_subtype_base,
less_than_wf,
eqmod_wf,
quotient-member-eq,
eqmod_equiv_rel,
modulus-equal-iff-eqmod
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
Error :equalityIsType1,
Error :universeIsType,
intEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
imageElimination,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairEquality,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
pointwiseFunctionalityForEquality,
functionEquality,
pertypeElimination,
Error :lambdaFormation_alt,
baseApply,
closedConclusion,
Error :equalityIsType4,
dependent_functionElimination,
Error :productIsType
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbZ{}\_n]. uiff((x mod n) = (y mod n);x = y)
Date html generated:
2019_06_20-PM-02_27_43
Last ObjectModification:
2018_10_15-PM-05_54_10
Theory : num_thy_1
Home
Index