Nuprl Lemma : multiply_assoc_int_mod
∀[n:ℤ]. ∀[x,y,z:ℤ_n]. (((x * y) * z) = (x * y * z) ∈ ℤ_n)
Proof
Definitions occuring in Statement :
int_mod: ℤ_n,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_mod: ℤ_n,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
eqmod_wf,
int_mod_wf,
istype-int,
quotient-member-eq,
eqmod_equiv_rel,
mul_assoc,
iff_weakening_equal,
eqmod_refl,
eqmod_functionality_wrt_eqmod,
multiply_functionality_wrt_eqmod,
eqmod_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
because_Cache,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
thin,
productElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
rename,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
productIsType,
sqequalBase,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
intEquality,
lambdaEquality_alt,
independent_isectElimination,
multiplyEquality,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[x,y,z:\mBbbZ{}\_n]. (((x * y) * z) = (x * y * z))
Date html generated:
2020_05_19-PM-10_03_00
Last ObjectModification:
2020_01_01-AM-10_06_59
Theory : num_thy_1
Home
Index