Nuprl Lemma : i-member-convex'
∀I:Interval. ∀a,b:ℝ.
((a ∈ I) ⇒ (b ∈ I) ⇒ (∀z:{z:ℝ| r0 < z} . ∀x,y:{z:ℝ| r0 ≤ z} . (((x + y) = z) ⇒ (((x * a) + (y * b)/z) ∈ I))))
Proof
Definitions occuring in Statement :
i-member: r ∈ I,
interval: Interval,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
iff: P ⇐⇒ Q
Lemmas referenced :
i-member-convex,
rdiv_wf,
sq_stable__rless,
int-to-real_wf,
rless_wf,
req_wf,
radd_wf,
rleq_wf,
i-member_wf,
real_wf,
interval_wf,
rmul_preserves_rleq,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
rinv_wf2,
sq_stable__rleq,
rleq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
trivial-rleq-radd,
req_inversion,
rmul_preserves_req,
rsub_wf,
rminus_wf,
itermAdd_wf,
itermMinus_wf,
req_functionality,
radd_functionality,
rminus_functionality,
real_term_value_add_lemma,
real_term_value_minus_lemma,
req-implies-req,
i-member_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
setElimination,
rename,
because_Cache,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
inhabitedIsType,
setIsType,
productElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
closedConclusion
Latex:
\mforall{}I:Interval. \mforall{}a,b:\mBbbR{}.
((a \mmember{} I)
{}\mRightarrow{} (b \mmember{} I)
{}\mRightarrow{} (\mforall{}z:\{z:\mBbbR{}| r0 < z\} . \mforall{}x,y:\{z:\mBbbR{}| r0 \mleq{} z\} . (((x + y) = z) {}\mRightarrow{} (((x * a) + (y * b)/z) \mmember{} I))))
Date html generated:
2019_10_29-AM-10_46_52
Last ObjectModification:
2019_01_08-PM-06_10_39
Theory : reals
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