Nuprl Lemma : rmul_functionality_wrt_rleq2
∀[x,y,z,w:ℝ]. ((x * w) ≤ (z * y)) supposing ((w ≤ y) and (x ≤ z) and (((r0 ≤ x) ∧ (r0 ≤ y)) ∨ ((r0 ≤ w) ∧ (r0 ≤ z))))
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,
or: P ∨ Q,
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,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
guard: {T}
Lemmas referenced :
less_than'_wf,
rsub_wf,
rmul_wf,
real_wf,
nat_plus_wf,
rleq_wf,
or_wf,
int-to-real_wf,
rmul_functionality_wrt_rleq,
rleq-implies-rleq,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rleq_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
unionElimination,
thin,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
productEquality,
voidElimination,
independent_isectElimination,
comment,
computeAll,
int_eqEquality,
intEquality,
voidEquality
Latex:
\mforall{}[x,y,z,w:\mBbbR{}].
((x * w) \mleq{} (z * y)) supposing
((w \mleq{} y) and
(x \mleq{} z) and
(((r0 \mleq{} x) \mwedge{} (r0 \mleq{} y)) \mvee{} ((r0 \mleq{} w) \mwedge{} (r0 \mleq{} z))))
Date html generated:
2017_10_03-AM-08_26_26
Last ObjectModification:
2017_07_28-AM-07_24_19
Theory : reals
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