Nuprl Lemma : rmul_reverses_rless
∀x,y,z:ℝ. ((x < z) ⇒ (y < r0) ⇒ ((z * y) < (x * y)))
Proof
Definitions occuring in Statement :
rless: x < y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
false: False,
not: ¬A,
iff: P ⇐⇒ Q
Lemmas referenced :
rless_wf,
int-to-real_wf,
real_wf,
rminus-reverses-rless,
rminus_wf,
rmul_wf,
rless_functionality,
real_term_polynomial,
itermSubtract_wf,
itermMinus_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
req_transitivity,
itermVar_wf,
itermMultiply_wf,
real_term_value_var_lemma,
real_term_value_mul_lemma,
req_inversion,
rminus-as-rmul,
rmul_functionality_wrt_rless,
rless-implies-rless,
rsub_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
minusEquality,
independent_isectElimination,
sqequalRule,
computeAll,
lambdaEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
int_eqEquality
Latex:
\mforall{}x,y,z:\mBbbR{}. ((x < z) {}\mRightarrow{} (y < r0) {}\mRightarrow{} ((z * y) < (x * y)))
Date html generated:
2017_10_03-AM-08_27_22
Last ObjectModification:
2017_07_28-AM-07_24_48
Theory : reals
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