Nuprl Lemma : rabs-diff-rmul
∀[a,b,c,d,x,y:ℝ]. ((|a - b| ≤ x) ⇒ (|c - d| ≤ y) ⇒ (|(a * c) - b * d| ≤ ((|a| * y) + (|d| * x))))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
uiff: uiff(P;Q),
uimplies: b supposing a,
req_int_terms: t1 ≡ t2,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T}
Lemmas referenced :
rleq_wf,
rabs_wf,
rsub_wf,
less_than'_wf,
radd_wf,
rmul_wf,
real_wf,
nat_plus_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermAdd_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
uimplies_transitivity,
rleq_functionality,
radd_functionality,
rabs-rmul,
req_weakening,
rleq_functionality_wrt_implies,
r-triangle-inequality,
rleq_weakening_equal,
rabs_functionality,
rmul_preserves_rleq2,
zero-rleq-rabs,
rminus_wf,
itermMinus_wf,
rmul_comm,
rmul_functionality,
real_term_value_minus_lemma,
rleq_weakening,
req_transitivity,
radd_functionality_wrt_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality,
independent_functionElimination,
promote_hyp
Latex:
\mforall{}[a,b,c,d,x,y:\mBbbR{}]. ((|a - b| \mleq{} x) {}\mRightarrow{} (|c - d| \mleq{} y) {}\mRightarrow{} (|(a * c) - b * d| \mleq{} ((|a| * y) + (|d| * x))))
Date html generated:
2018_05_22-PM-01_59_47
Last ObjectModification:
2017_10_25-AM-11_07_31
Theory : reals
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