Nuprl Lemma : chrem_exists_a
∀r:ℕ+. ∀s:{s':ℕ+| CoPrime(r,s')} . ∀a,b:ℤ. (∃x:ℤ [((x ≡ a mod r) ∧ (x ≡ b 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]} ,
int: ℤ
Definitions unfolded in proof :
prop: ℙ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
coprime: CoPrime(a,b),
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
top: Top
Lemmas referenced :
nat_plus_wf,
coprime_wf,
istype-int,
sq_stable__coprime,
chrem_exists_aux_a,
gcd_p_sym,
eqmod_wf,
add-zero,
zero-mul,
one-mul,
mul-commutes,
multiply_functionality_wrt_eqmod,
add_functionality_wrt_eqmod,
eqmod_functionality_wrt_eqmod,
eqmod_weakening,
istype-void,
zero-add
Rules used in proof :
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
Error :universeIsType,
Error :setIsType,
hypothesis,
extract_by_obid,
introduction,
cut,
hypothesisEquality,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
Error :productIsType,
independent_pairFormation,
productElimination,
multiplyEquality,
addEquality,
Error :dependent_set_memberFormation_alt,
independent_isectElimination,
voidElimination,
Error :isect_memberEquality_alt,
because_Cache
Latex:
\mforall{}r:\mBbbN{}\msupplus{}. \mforall{}s:\{s':\mBbbN{}\msupplus{}| CoPrime(r,s')\} . \mforall{}a,b:\mBbbZ{}. (\mexists{}x:\mBbbZ{} [((x \mequiv{} a mod r) \mwedge{} (x \mequiv{} b mod s))])
Date html generated:
2019_06_20-PM-02_25_06
Last ObjectModification:
2019_01_17-AM-09_06_59
Theory : num_thy_1
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