Nuprl Lemma : ratsub_wf
∀[a,b:ℤ × ℕ+]. (ratsub(a;b) ∈ {r:ℤ × ℕ+| ratreal(r) = (ratreal(a) - ratreal(b))} )
Proof
Definitions occuring in Statement :
ratsub: ratsub(x;y),
ratreal: ratreal(r),
rsub: x - y,
req: x = y,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ratsub: ratsub(x;y),
subtype_rel: A ⊆r B,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top
Lemmas referenced :
istype-int,
nat_plus_wf,
ratadd_wf,
int-rat-mul_wf,
req_wf,
ratreal_wf,
rsub_wf,
radd_wf,
rmul_wf,
int-to-real_wf,
int-rmul_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_functionality,
req_transitivity,
ratreal-ratadd,
radd_functionality,
ratreal-int-rat-mul,
int-rmul-req,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
hypothesisEquality,
isect_memberEquality_alt,
isectElimination,
thin,
isectIsTypeImplies,
productIsType,
extract_by_obid,
universeIsType,
dependent_set_memberEquality_alt,
minusEquality,
natural_numberEquality,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
because_Cache,
productElimination,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
int_eqEquality,
voidElimination
Latex:
\mforall{}[a,b:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}]. (ratsub(a;b) \mmember{} \{r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(r) = (ratreal(a) - ratreal(b))\} )
Date html generated:
2019_10_30-AM-09_25_22
Last ObjectModification:
2019_01_11-PM-00_22_42
Theory : reals
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