Nuprl Lemma : rmul_functionality_wrt_rless
∀x,y,z:ℝ. ((x < z) ⇒ (r0 < y) ⇒ ((x * y) < (z * 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,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
uimplies: b supposing a
Lemmas referenced :
rlessw_wf,
rmul_wf,
rless-iff-rpositive,
int-to-real_wf,
rless_wf,
real_wf,
rpositive-rmul,
rsub_wf,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_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,
rpositive_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
productElimination,
independent_functionElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination
Latex:
\mforall{}x,y,z:\mBbbR{}. ((x < z) {}\mRightarrow{} (r0 < y) {}\mRightarrow{} ((x * y) < (z * y)))
Date html generated:
2017_10_03-AM-08_26_38
Last ObjectModification:
2017_07_28-AM-07_24_27
Theory : reals
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