Nuprl Lemma : fractions-rless
∀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),
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 :
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
uiff: uiff(P;Q)
Lemmas referenced :
real_wf,
rmul_wf,
int-to-real_wf,
rdiv_wf,
rless_wf,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv,
req_weakening,
rmul_functionality,
req_transitivity,
rless_functionality,
itermConstant_wf,
rmul-one,
req-iff-rsub-is-0,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rmul_preserves_rless
Rules used in proof :
natural_numberEquality,
inrFormation,
hypothesis,
sqequalRule,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
approximateComputation,
because_Cache,
productElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}a,b,c,d:\mBbbR{}. ((r0 < c) {}\mRightarrow{} (r0 < d) {}\mRightarrow{} ((a/c) < (b/d) \mLeftarrow{}{}\mRightarrow{} (a * d) < (b * c)))
Date html generated:
2017_10_03-AM-08_38_55
Last ObjectModification:
2017_07_29-PM-08_39_25
Theory : reals
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