Nuprl Lemma : rabs-difference-is-zero
∀x,y:ℝ. (|x - y| = r0 ⇐⇒ x = y)
Proof
Definitions occuring in Statement :
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
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,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
absval: |i|
Lemmas referenced :
req_wf,
rabs_wf,
rsub_wf,
int-to-real_wf,
real_wf,
rabs-difference-bound-rleq,
rleq_weakening,
rleq_antisymmetry,
rleq-implies-rleq,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
req-int,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
req_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
req_transitivity,
rabs-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
inhabitedIsType,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
because_Cache,
sqequalRule,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
minusEquality,
unionElimination,
dependent_pairFormation_alt
Latex:
\mforall{}x,y:\mBbbR{}. (|x - y| = r0 \mLeftarrow{}{}\mRightarrow{} x = y)
Date html generated:
2019_10_29-AM-10_00_19
Last ObjectModification:
2019_05_04-AM-11_14_07
Theory : reals
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