Nuprl Lemma : eqmod-divides-implies
∀m,m':ℤ. ((m' | m) ⇒ {∀a,b:ℤ. ((a ≡ b mod m) ⇒ (a ≡ b mod m'))})
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
divides: b | a,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
eqmod: a ≡ b mod m,
divides: b | a,
exists: ∃x:A. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
eqmod_wf,
divides_wf,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
mul_assoc,
iff_weakening_equal,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
intEquality,
productElimination,
promote_hyp,
instantiate,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
multiplyEquality,
baseApply,
closedConclusion
Latex:
\mforall{}m,m':\mBbbZ{}. ((m' | m) {}\mRightarrow{} \{\mforall{}a,b:\mBbbZ{}. ((a \mequiv{} b mod m) {}\mRightarrow{} (a \mequiv{} b mod m'))\})
Date html generated:
2017_04_17-AM-09_42_52
Last ObjectModification:
2017_02_27-PM-05_37_26
Theory : num_thy_1
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