Nuprl Lemma : rem_mul2
∀[x,y:ℕ]. ∀[m:ℕ+]. ((x * y rem m) = ((x rem m) * y rem m) ∈ ℤ)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
remainder: n rem m,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
true: True,
nat: ℕ,
subtype_rel: A ⊆r B,
squash: ↓T,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
nat_plus_wf,
istype-nat,
remainder_wf,
remainder_wfa,
nat_plus_inc_int_nzero,
equal_wf,
rem_mul,
iff_weakening_equal,
rem_rem_to_rem
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
hypothesis,
Error :universeIsType,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
Error :isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
intEquality,
natural_numberEquality,
because_Cache,
multiplyEquality,
setElimination,
rename,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[x,y:\mBbbN{}]. \mforall{}[m:\mBbbN{}\msupplus{}]. ((x * y rem m) = ((x rem m) * y rem m))
Date html generated:
2019_06_20-PM-02_32_06
Last ObjectModification:
2019_03_06-AM-10_53_44
Theory : num_thy_1
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