Nuprl Lemma : rmul-nonneg-rabs
∀[x,y:ℝ]. (x * |y|) = |x * y| supposing r0 ≤ x
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rmul_wf,
rabs_wf,
rleq_wf,
int-to-real_wf,
real_wf,
req_weakening,
req_functionality,
rabs-rmul,
rmul_functionality,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
universeIsType,
natural_numberEquality,
sqequalRule,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
independent_isectElimination,
productElimination
Latex:
\mforall{}[x,y:\mBbbR{}]. (x * |y|) = |x * y| supposing r0 \mleq{} x
Date html generated:
2019_10_29-AM-09_39_09
Last ObjectModification:
2019_02_13-PM-02_32_11
Theory : reals
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