Nuprl Lemma : functions-equal-on-rationals
∀I:Interval
((∃u,v:{a:ℝ| a ∈ I} . u ≠ v)
⇒ (∀f,g:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (g(x) = g(y))))
⇒ (∀z:ℤ. ∀d:ℕ+. (((r(z)/r(d)) ∈ I) ⇒ (f((r(z)/r(d))) = g((r(z)/r(d))))))
⇒ (∀a:{x:ℝ| x ∈ I} . (f(a) = g(a))))))
Proof
Definitions occuring in Statement :
r-ap: f(x),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
int: ℤ
Definitions unfolded in proof :
so_apply: x[s],
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
interval_wf,
rneq_wf,
rfun_wf,
r-ap_wf,
all_wf,
set_wf,
rationals-dense-in-interval,
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
req_wf,
nat_plus_wf,
exists_wf,
real_wf,
i-member_wf,
functions-equal-on-dense,
i-member_functionality,
sq_stable__i-member,
req_transitivity,
req_inversion
Rules used in proof :
setEquality,
functionEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
natural_numberEquality,
independent_functionElimination,
productElimination,
inrFormation,
independent_isectElimination,
sqequalRule,
because_Cache,
rename,
setElimination,
hypothesis,
lambdaEquality,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
baseClosed,
imageMemberEquality,
dependent_set_memberEquality
Latex:
\mforall{}I:Interval
((\mexists{}u,v:\{a:\mBbbR{}| a \mmember{} I\} . u \mneq{} v)
{}\mRightarrow{} (\mforall{}f,g:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f(x) = f(y))))
{}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (g(x) = g(y))))
{}\mRightarrow{} (\mforall{}z:\mBbbZ{}. \mforall{}d:\mBbbN{}\msupplus{}. (((r(z)/r(d)) \mmember{} I) {}\mRightarrow{} (f((r(z)/r(d))) = g((r(z)/r(d))))))
{}\mRightarrow{} (\mforall{}a:\{x:\mBbbR{}| x \mmember{} I\} . (f(a) = g(a))))))
Date html generated:
2017_10_03-AM-10_21_20
Last ObjectModification:
2017_07_31-PM-00_08_29
Theory : reals
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