Nuprl Lemma : rmul_functionality_wrt_rless2
∀x,y,z,w:ℝ. ((r0 < w) ⇒ (x < z) ⇒ (x * w) < (z * y) supposing w ≤ y supposing r0 ≤ z)
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
guard: {T}
Lemmas referenced :
less_than'_wf,
rsub_wf,
int-to-real_wf,
real_wf,
nat_plus_wf,
rmul_functionality_wrt_rless,
rmul_functionality_wrt_rleq,
rleq_wf,
rless_wf,
rless-implies-rless,
rmul_wf,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_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,
rleq-implies-rleq,
rless_transitivity1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
extract_by_obid,
isectElimination,
applyEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}x,y,z,w:\mBbbR{}. ((r0 < w) {}\mRightarrow{} (x < z) {}\mRightarrow{} (x * w) < (z * y) supposing w \mleq{} y supposing r0 \mleq{} z)
Date html generated:
2017_10_03-AM-08_26_44
Last ObjectModification:
2017_07_28-AM-07_24_32
Theory : reals
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