Nuprl Lemma : infinitesmal-iff
∀x:ℝ*. (is-infinitesmal(x) ⇐⇒ ∀r:{r:ℝ| r0 < r} . |x| < (r)*)
Proof
Definitions occuring in Statement :
is-infinitesmal: is-infinitesmal(x),
rstar: (x)*,
rless*: x < y,
rabs*: |x|,
real*: ℝ*,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
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],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
is-infinitesmal: is-infinitesmal(x),
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top
Lemmas referenced :
set_wf,
real_wf,
rless_wf,
int-to-real_wf,
is-infinitesmal_wf,
all_wf,
rless*_wf,
rabs*_wf,
rstar_wf,
real*_wf,
small-reciprocal-real,
rstar-rless,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless*_transitivity2,
rleq*_weakening_rless,
nat_plus_wf,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
setEquality,
setElimination,
rename,
dependent_functionElimination,
productElimination,
because_Cache,
independent_isectElimination,
inrFormation,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
multiplyEquality,
dependent_set_memberEquality
Latex:
\mforall{}x:\mBbbR{}*. (is-infinitesmal(x) \mLeftarrow{}{}\mRightarrow{} \mforall{}r:\{r:\mBbbR{}| r0 < r\} . |x| < (r)*)
Date html generated:
2018_05_22-PM-09_29_19
Last ObjectModification:
2017_10_06-PM-05_27_25
Theory : reals_2
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