Nuprl Lemma : add_reduce_eqmod
∀m,x,y:ℤ. ((x + y) ≡ x mod m ⇐⇒ y ≡ 0 mod m)
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
top: Top,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
eqmod_wf,
eqmod_weakening,
minus-one-mul,
add-swap,
add-associates,
add-mul-special,
zero-mul,
zero-add,
add-commutes,
subtype_base_sq,
int_subtype_base,
add_functionality_wrt_eqmod,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
hypothesis,
natural_numberEquality,
intEquality,
minusEquality,
dependent_functionElimination,
independent_isectElimination,
sqequalRule,
multiplyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
instantiate,
cumulativity,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}m,x,y:\mBbbZ{}. ((x + y) \mequiv{} x mod m \mLeftarrow{}{}\mRightarrow{} y \mequiv{} 0 mod m)
Date html generated:
2016_10_21-AM-11_09_02
Last ObjectModification:
2016_07_12-AM-06_01_35
Theory : num_thy_1
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