Nuprl Lemma : quadratic-residue_functionality
∀a,a',p:ℤ. ((a' ≡ a mod p) ⇒ (a' is a quadratic residue mod p ⇐⇒ a is a quadratic residue mod p))
Proof
Definitions occuring in Statement :
quadratic-residue: a is a quadratic residue mod p,
eqmod: a ≡ b mod m,
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,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
quadratic-residue: a is a quadratic residue mod p,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
eqmod_wf,
exists_wf,
iff_wf,
eqmod_functionality_wrt_eqmod,
eqmod_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
intEquality,
independent_pairFormation,
sqequalRule,
lambdaEquality,
multiplyEquality,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
existsFunctionality,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
existsLevelFunctionality
Latex:
\mforall{}a,a',p:\mBbbZ{}. ((a' \mequiv{} a mod p) {}\mRightarrow{} (a' is a quadratic residue mod p \mLeftarrow{}{}\mRightarrow{} a is a quadratic residue mod p))
Date html generated:
2016_05_14-PM-04_27_27
Last ObjectModification:
2015_12_26-PM-08_05_24
Theory : num_thy_1
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