Nuprl Lemma : rabs-rmul-rleq-rabs
∀[x,y,a,b:ℝ]. (|x * y| ≤ |a * b|) supposing ((|y| ≤ |b|) and (|x| ≤ |a|))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rmul: a * b,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
less_than'_wf,
rsub_wf,
rabs_wf,
rmul_wf,
real_wf,
nat_plus_wf,
rleq_wf,
rabs-rmul-rleq,
rleq_functionality,
req_weakening,
rabs-rmul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_isectElimination
Latex:
\mforall{}[x,y,a,b:\mBbbR{}]. (|x * y| \mleq{} |a * b|) supposing ((|y| \mleq{} |b|) and (|x| \mleq{} |a|))
Date html generated:
2016_05_18-AM-07_14_45
Last ObjectModification:
2015_12_28-AM-00_42_23
Theory : reals
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