Nuprl Lemma : int-has-rational-square-root
∀n:ℤ. (∃q:ℚ. ((q * q) = n ∈ ℚ) ⇐⇒ ∃m:ℤ. ((m * m) = n ∈ ℤ))
Proof
Definitions occuring in Statement :
qmul: r * s,
rationals: ℚ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
exists_wf,
rationals_wf,
equal_wf,
qmul_wf,
int-subtype-rationals,
int-with-rational-square-root,
equal_functionality_wrt_subtype_rel2,
qmul-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
intEquality,
multiplyEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination
Latex:
\mforall{}n:\mBbbZ{}. (\mexists{}q:\mBbbQ{}. ((q * q) = n) \mLeftarrow{}{}\mRightarrow{} \mexists{}m:\mBbbZ{}. ((m * m) = n))
Date html generated:
2016_05_15-PM-11_44_27
Last ObjectModification:
2015_12_27-PM-07_24_57
Theory : rationals
Home
Index