Nuprl Lemma : eqmod-eqmod-div
∀m,m',a,a',b,b':ℤ. ((m' | m) ⇒ (a ≡ a' mod m) ⇒ (b ≡ b' mod m) ⇒ (a ≡ b mod m' ⇐⇒ a' ≡ b' mod m'))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
divides: b | a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
eqmod: a ≡ b mod m,
divides: b | a,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
mul_assoc,
iff_weakening_equal,
eqmod_wf,
divides_wf,
subtract_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermMultiply_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
promote_hyp,
cut,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
addEquality,
multiplyEquality,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
baseApply,
closedConclusion
Latex:
\mforall{}m,m',a,a',b,b':\mBbbZ{}.
((m' | m) {}\mRightarrow{} (a \mequiv{} a' mod m) {}\mRightarrow{} (b \mequiv{} b' mod m) {}\mRightarrow{} (a \mequiv{} b mod m' \mLeftarrow{}{}\mRightarrow{} a' \mequiv{} b' mod m'))
Date html generated:
2017_04_17-AM-09_42_49
Last ObjectModification:
2017_02_27-PM-05_37_40
Theory : num_thy_1
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