Nuprl Lemma : rmul_preserves_rleq2
∀[x,y,z:ℝ]. ((x * y) ≤ (z * y)) supposing ((x ≤ z) and (r0 ≤ y))
Proof
Definitions occuring in Statement :
rleq: 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,
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
Lemmas referenced :
less_than'_wf,
rsub_wf,
rmul_wf,
real_wf,
nat_plus_wf,
rleq_wf,
int-to-real_wf,
rmul_functionality_wrt_rleq
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,z:\mBbbR{}]. ((x * y) \mleq{} (z * y)) supposing ((x \mleq{} z) and (r0 \mleq{} y))
Date html generated:
2016_05_18-AM-07_12_35
Last ObjectModification:
2015_12_28-AM-00_41_24
Theory : reals
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