Nuprl Lemma : chrem_exists_aux_a
∀r:ℕ+. ∀s:{s':ℕ+| CoPrime(r,s')} . (∃x:ℤ [((x ≡ 1 mod r) ∧ (x ≡ 0 mod s))])
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
coprime: CoPrime(a,b),
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
prop: ℙ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
top: Top,
eqmod: a ≡ b mod m,
subtract: n - m,
divides: b | a,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
subtype_rel: A ⊆r B
Lemmas referenced :
coprime_wf,
nat_plus_wf,
sq_stable__coprime,
coprime_bezout_id,
eqmod_wf,
istype-void,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
add-mul-special,
zero-mul,
add-zero,
minus-zero,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :setIsType,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
Error :dependent_set_memberFormation_alt,
multiplyEquality,
Error :productIsType,
natural_numberEquality,
because_Cache,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
productEquality,
Error :isect_memberEquality_alt,
voidElimination,
minusEquality,
independent_pairFormation,
Error :dependent_pairFormation_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
Error :equalityIsType4,
equalityTransitivity,
applyEquality
Latex:
\mforall{}r:\mBbbN{}\msupplus{}. \mforall{}s:\{s':\mBbbN{}\msupplus{}| CoPrime(r,s')\} . (\mexists{}x:\mBbbZ{} [((x \mequiv{} 1 mod r) \mwedge{} (x \mequiv{} 0 mod s))])
Date html generated:
2019_06_20-PM-02_24_56
Last ObjectModification:
2018_10_03-AM-00_13_17
Theory : num_thy_1
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