Nuprl Lemma : rabs-diff-rdiv
∀[a,b,c,d,x,y:ℝ].
(c ≠ r0 ⇒ d ≠ r0 ⇒ (|a - b| ≤ x) ⇒ (|(r1/c) - (r1/d)| ≤ y) ⇒ (|(a/c) - (b/d)| ≤ ((|a| * y) + (|(r1/d)| * x))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rabs-diff-rmul,
rdiv_wf,
int-to-real_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rneq_wf,
le_witness_for_triv,
real_wf,
rmul_wf,
radd_wf,
rminus_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermAdd_wf,
itermMinus_wf,
itermConstant_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_functionality,
req_transitivity,
rabs_functionality,
radd_functionality,
rminus-rdiv,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
independent_isectElimination,
independent_functionElimination,
universeIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
because_Cache,
approximateComputation,
int_eqEquality,
voidElimination
Latex:
\mforall{}[a,b,c,d,x,y:\mBbbR{}].
(c \mneq{} r0
{}\mRightarrow{} d \mneq{} r0
{}\mRightarrow{} (|a - b| \mleq{} x)
{}\mRightarrow{} (|(r1/c) - (r1/d)| \mleq{} y)
{}\mRightarrow{} (|(a/c) - (b/d)| \mleq{} ((|a| * y) + (|(r1/d)| * x))))
Date html generated:
2019_10_29-AM-10_21_50
Last ObjectModification:
2019_07_05-PM-00_18_35
Theory : reals
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