Nuprl Lemma : convex-comb_wf
∀[I:Interval]. ∀[x,y:{x:ℝ| x ∈ I} ]. ∀[r:{r:ℝ| r0 ≤ r} ]. ∀[s:{s:ℝ| (r0 ≤ s) ∧ (r0 < (r + s))} ].
(convex-comb(x;y;r;s) ∈ {x:ℝ| x ∈ I} )
Proof
Definitions occuring in Statement :
convex-comb: convex-comb(x;y;r;s),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
convex-comb: convex-comb(x;y;r;s),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
true: True,
real: ℝ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
cand: A c∧ B,
i-member: r ∈ I,
interval: Interval,
rge: x ≥ y
Lemmas referenced :
interval_wf,
rleq_wf,
real_wf,
set_wf,
i-member_wf,
int-to-real_wf,
rless_wf,
rmul_wf,
radd_wf,
rdiv_wf,
rless_functionality,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv3,
radd_functionality,
req_transitivity,
rleq_functionality,
nat_plus_wf,
regular-int-seq_wf,
rinv_wf2,
req-iff-rsub-is-0,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rmul_preserves_rless,
rmul_preserves_rleq,
radd_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rsub_wf,
rleq-implies-rleq,
rmul_preserves_rleq2,
radd-positive-implies,
rmul_comm,
rleq_weakening_rless,
rless-implies-rless,
rless_functionality_wrt_implies,
radd_comm,
radd-preserves-rless,
radd_functionality_wrt_rless1,
radd_functionality_wrt_rless2
Rules used in proof :
isect_memberEquality,
productEquality,
lambdaEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
natural_numberEquality,
inrFormation,
independent_isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
sqequalRule,
productElimination,
sqequalHypSubstitution,
dependent_set_memberEquality,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
approximateComputation,
applyEquality,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
unionElimination
Latex:
\mforall{}[I:Interval]. \mforall{}[x,y:\{x:\mBbbR{}| x \mmember{} I\} ]. \mforall{}[r:\{r:\mBbbR{}| r0 \mleq{} r\} ]. \mforall{}[s:\{s:\mBbbR{}| (r0 \mleq{} s) \mwedge{} (r0 < (r + s))\} ].
(convex-comb(x;y;r;s) \mmember{} \{x:\mBbbR{}| x \mmember{} I\} )
Date html generated:
2017_10_04-PM-11_11_56
Last ObjectModification:
2017_07_29-PM-06_22_07
Theory : reals_2
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