Nuprl Lemma : req-iff-rabs-rleq
∀x,y:ℝ. (x = y ⇐⇒ ∀m:ℕ+. (|x - y| ≤ (r1/r(m))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
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],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_apply: x[s],
subtype_rel: A ⊆r B,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
absval: |i|,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
squash: ↓T,
sq_exists: ∃x:{A| B[x]},
rless: x < y
Lemmas referenced :
nat_plus_wf,
req_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
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_wf,
real_wf,
absval_wf,
rinv_wf2,
rmul_wf,
rleq-int-fractions2,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
uiff_transitivity2,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_transitivity,
real_term_value_mul_lemma,
rinv-as-rdiv,
squash_wf,
true_wf,
rabs-int,
rless_transitivity1,
rless_irreflexivity,
small-reciprocal-real,
req-iff-not-rneq,
rneq-iff-rabs,
rneq_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality,
minusEquality,
multiplyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
impliesFunctionality,
addLevel,
lemma_by_obid,
dependent_set_memberEquality
Latex:
\mforall{}x,y:\mBbbR{}. (x = y \mLeftarrow{}{}\mRightarrow{} \mforall{}m:\mBbbN{}\msupplus{}. (|x - y| \mleq{} (r1/r(m))))
Date html generated:
2017_10_03-AM-09_06_12
Last ObjectModification:
2017_07_28-AM-07_41_59
Theory : reals
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