Nuprl Lemma : rmul_preserves_rleq3
∀[x,y,a,b:ℝ]. ((x * y) ≤ (a * b)) supposing ((x ≤ a) and (y ≤ b) and ((r0 ≤ x) ∧ (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],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
prop: ℙ,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
rmul_preserves_rleq2,
rleq_transitivity,
int-to-real_wf,
le_witness_for_triv,
rleq_wf,
real_wf,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
natural_numberEquality,
promote_hyp,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
productIsType,
because_Cache,
approximateComputation,
int_eqEquality,
voidElimination
Latex:
\mforall{}[x,y,a,b:\mBbbR{}]. ((x * y) \mleq{} (a * b)) supposing ((x \mleq{} a) and (y \mleq{} b) and ((r0 \mleq{} x) \mwedge{} (r0 \mleq{} y)))
Date html generated:
2019_10_29-AM-09_38_11
Last ObjectModification:
2019_02_04-AM-10_02_05
Theory : reals
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