Nuprl Lemma : ravg-between
∀x,y:ℝ. ((x < y) ⇒ ((x < ravg(x;y)) ∧ (ravg(x;y) < y)))
Proof
Definitions occuring in Statement :
ravg: ravg(x;y),
rless: x < y,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
ravg: ravg(x;y),
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
rdiv: (x/y)
Lemmas referenced :
rmul_preserves_rless,
rdiv_wf,
rless-int,
rless_wf,
real_wf,
rmul_wf,
int-to-real_wf,
radd_wf,
rinv_wf2,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
req-iff-rsub-is-0,
rsub_wf,
rless_functionality,
req_transitivity,
radd_functionality,
rmul_functionality,
rmul-rinv,
req_weakening,
rmul-identity1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
isectElimination,
independent_isectElimination,
sqequalRule,
hypothesis,
inrFormation,
productElimination,
independent_functionElimination,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} ((x < ravg(x;y)) \mwedge{} (ravg(x;y) < y)))
Date html generated:
2017_10_03-AM-08_41_52
Last ObjectModification:
2017_07_28-AM-07_34_35
Theory : reals
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