Nuprl Lemma : rmul-nonneg
∀[x,y:ℝ]. r0 ≤ (x * y) supposing ((r0 ≤ x) ∧ (r0 ≤ y)) ∨ ((x ≤ r0) ∧ (y ≤ r0))
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],
or: P ∨ Q,
and: P ∧ Q,
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,
or: P ∨ Q,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmul_functionality,
rmul-assoc,
req_inversion,
rminus-zero,
rminus-rminus,
rmul-one-both,
rminus_functionality,
rmul_over_rminus,
req_transitivity,
rmul-minus,
rmul-int,
rless-int,
rmul_reverses_rleq_iff,
rminus_wf,
rmul_comm,
req_weakening,
rmul-zero-both,
rleq_functionality,
uiff_transitivity,
rmul_preserves_rleq2,
rleq_wf,
and_wf,
or_wf,
nat_plus_wf,
real_wf,
int-to-real_wf,
rmul_wf,
rsub_wf,
less_than'_wf
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,
natural_numberEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
unionElimination,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
multiplyEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. r0 \mleq{} (x * y) supposing ((r0 \mleq{} x) \mwedge{} (r0 \mleq{} y)) \mvee{} ((x \mleq{} r0) \mwedge{} (y \mleq{} r0))
Date html generated:
2016_05_18-AM-07_33_32
Last ObjectModification:
2016_01_17-AM-02_01_20
Theory : reals
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