Nuprl Lemma : convex-comb-homog
∀[x,y,r:ℝ]. ∀[s:{s:ℝ| r + s ≠ r0} ]. ∀[t:{t:ℝ| t ≠ r0} ]. (convex-comb(x;y;r * t;s * t) = convex-comb(x;y;r;s))
Proof
Definitions occuring in Statement :
convex-comb: convex-comb(x;y;r;s),
rneq: x ≠ y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2
Lemmas referenced :
convex-comb-req,
req_functionality,
rdiv_wf,
rsub_wf,
real_wf,
set_wf,
rneq_wf,
rmul-zero-both,
rmul-distrib2,
rneq_functionality,
rmul_preserves_rneq_iff2,
int-to-real_wf,
radd_wf,
sq_stable_rneq,
rmul_wf,
convex-comb_wf1,
sq_stable__req,
rsub_functionality,
rmul_functionality,
radd_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,
req_weakening,
req-iff-rsub-is-0,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
fractions-req
Rules used in proof :
lambdaEquality,
dependent_set_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_isectElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
dependent_functionElimination,
hypothesis,
because_Cache,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation
Latex:
\mforall{}[x,y,r:\mBbbR{}]. \mforall{}[s:\{s:\mBbbR{}| r + s \mneq{} r0\} ]. \mforall{}[t:\{t:\mBbbR{}| t \mneq{} r0\} ].
(convex-comb(x;y;r * t;s * t) = convex-comb(x;y;r;s))
Date html generated:
2017_10_04-PM-11_12_17
Last ObjectModification:
2017_07_29-PM-08_21_09
Theory : reals_2
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