Nuprl Lemma : rmul_functionality_wrt_rleq
∀[x,y,z:ℝ]. ((x * y) ≤ (z * y)) supposing ((r0 ≤ y) and (x ≤ z))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
rleq: x ≤ y,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rnonneg-rmul,
rsub_wf,
int-to-real_wf,
less_than'_wf,
rmul_wf,
real_wf,
nat_plus_wf,
rnonneg_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,
rnonneg_functionality
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
lambdaEquality,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
computeAll,
int_eqEquality,
intEquality,
voidEquality,
independent_isectElimination
Latex:
\mforall{}[x,y,z:\mBbbR{}]. ((x * y) \mleq{} (z * y)) supposing ((r0 \mleq{} y) and (x \mleq{} z))
Date html generated:
2017_10_03-AM-08_26_19
Last ObjectModification:
2017_07_28-AM-07_24_15
Theory : reals
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