Nuprl Lemma : q-elim
∀r:ℚ. ∃p:ℤ. ∃q:ℕ+. ((¬(q = 0 ∈ ℚ)) c∧ (r = (p/q) ∈ ℚ))
Proof
Definitions occuring in Statement :
qdiv: (r/s),
rationals: ℚ,
nat_plus: ℕ+,
cand: A c∧ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
cand: A c∧ B,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
qdiv: (r/s),
qinv: 1/r,
qmul: r * s,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
exists: ∃x:A. B[x],
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
uiff: uiff(P;Q)
Lemmas referenced :
equals-qrep,
exists_wf,
nat_plus_wf,
not_wf,
equal-wf-T-base,
subtype_rel_set,
rationals_wf,
less_than_wf,
int-subtype-rationals,
equal_wf,
qdiv_wf,
qrep_wf,
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
set-valueall-type,
product-valueall-type,
mul-one,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
int-equal-in-rationals
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
equalitySymmetry,
hypothesis,
hyp_replacement,
applyLambdaEquality,
intEquality,
sqequalRule,
lambdaEquality,
productEquality,
because_Cache,
applyEquality,
natural_numberEquality,
independent_isectElimination,
baseClosed,
productElimination,
equalityTransitivity,
dependent_functionElimination,
independent_functionElimination,
callbyvalueReduce,
isintReduceTrue,
setElimination,
rename,
independent_pairEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addLevel,
impliesFunctionality
Latex:
\mforall{}r:\mBbbQ{}. \mexists{}p:\mBbbZ{}. \mexists{}q:\mBbbN{}\msupplus{}. ((\mneg{}(q = 0)) c\mwedge{} (r = (p/q)))
Date html generated:
2018_05_21-PM-11_47_42
Last ObjectModification:
2017_07_26-PM-06_43_12
Theory : rationals
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