Nuprl Lemma : chrem_wf
∀r,s,a,b:ℤ. (chrem(a;r;b;s) ∈ {x:ℤ| (x ≡ a mod r) ∧ (x ≡ b mod s)} + (¬(∃x:ℤ [((x ≡ a mod r) ∧ (x ≡ b mod s))])))
Proof
Definitions occuring in Statement :
chrem: chrem(a;r;b;s),
eqmod: a ≡ b mod m,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
union: left + right,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
chrem: chrem(a;r;b;s),
genrec-ap: genrec-ap,
absval: |i|,
chinese-remainder2-extract,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
chinese-remainder2-extract,
subtype_rel_self,
decidable_wf,
sq_exists_wf,
eqmod_wf,
not_wf,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
intEquality,
lambdaEquality_alt,
productEquality,
hypothesisEquality,
inhabitedIsType,
unionEquality,
setEquality,
because_Cache
Latex:
\mforall{}r,s,a,b:\mBbbZ{}.
(chrem(a;r;b;s)
\mmember{} \{x:\mBbbZ{}| (x \mequiv{} a mod r) \mwedge{} (x \mequiv{} b mod s)\} + (\mneg{}(\mexists{}x:\mBbbZ{} [((x \mequiv{} a mod r) \mwedge{} (x \mequiv{} b mod s))])))
Date html generated:
2020_05_20-AM-08_13_45
Last ObjectModification:
2019_11_27-PM-03_15_47
Theory : general
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