Nuprl Lemma : rbetween-convex
∀x,a,b:ℝ. ((a < b) ⇒ a≤x≤b ⇒ (∃t:ℝ. ((r0 ≤ t) ∧ (t ≤ r1) ∧ (x = ((t * a) + ((r1 - t) * b))))))
Proof
Definitions occuring in Statement :
rbetween: x≤y≤z,
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rbetween: x≤y≤z,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
exists: ∃x:A. B[x],
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
prop: ℙ,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rdiv: (x/y)
Lemmas referenced :
rless-implies-rless,
int-to-real_wf,
rsub_wf,
rdiv_wf,
rless_wf,
rmul_preserves_rleq,
rmul_preserves_req,
radd_wf,
rmul_wf,
rleq_wf,
req_wf,
rbetween_wf,
real_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rminus_wf,
rmul-zero-both,
rinv_wf2,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
rleq-implies-rleq,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality,
req_transitivity,
radd_functionality,
rminus_functionality,
rmul-rinv3,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
req_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_isectElimination,
dependent_pairFormation,
because_Cache,
sqequalRule,
inrFormation,
independent_pairFormation,
productEquality,
independent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}x,a,b:\mBbbR{}. ((a < b) {}\mRightarrow{} a\mleq{}x\mleq{}b {}\mRightarrow{} (\mexists{}t:\mBbbR{}. ((r0 \mleq{} t) \mwedge{} (t \mleq{} r1) \mwedge{} (x = ((t * a) + ((r1 - t) * b))))))
Date html generated:
2018_05_22-PM-01_50_02
Last ObjectModification:
2017_10_20-PM-05_16_32
Theory : reals
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