Nuprl Lemma : fractions-rleq
∀a,b,c,d:ℝ. ((r0 < c) ⇒ (r0 < d) ⇒ ((a/c) ≤ (b/d) ⇐⇒ (a * d) ≤ (b * c)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rleq_wf,
rdiv_wf,
rless_wf,
int-to-real_wf,
rmul_wf,
real_wf,
rmul_preserves_rleq2,
rleq_weakening_rless,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
rmul-one,
itermConstant_wf,
rleq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul_preserves_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
hypothesis,
inrFormation,
natural_numberEquality,
because_Cache,
independent_functionElimination,
productElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}a,b,c,d:\mBbbR{}. ((r0 < c) {}\mRightarrow{} (r0 < d) {}\mRightarrow{} ((a/c) \mleq{} (b/d) \mLeftarrow{}{}\mRightarrow{} (a * d) \mleq{} (b * c)))
Date html generated:
2017_10_03-AM-08_39_07
Last ObjectModification:
2017_09_07-PM-06_37_36
Theory : reals
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