Nuprl Lemma : IVT-locally-non-constant
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
∀c:ℝ. ((f(a) ≤ c) ⇒ (c ≤ f(b)) ⇒ locally-non-constant(f;a;b;c) ⇒ (∃x:ℝ [((x ∈ [a, b]) ∧ (f(x) = c))]))
supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y]))
Proof
Definitions occuring in Statement :
locally-non-constant: locally-non-constant(f;a;b;c),
r-ap: f(x),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_apply: x[s],
rfun: I ⟶ℝ,
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
and: P ∧ Q,
i-member: r ∈ I,
rccint: [l, u],
prop: ℙ,
exists: ∃x:A. B[x],
cand: A c∧ B,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
false: False,
uiff: uiff(P;Q),
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
true: True,
rneq: x ≠ y,
sq_exists: ∃x:A [B[x]],
rleq: x ≤ y,
rnonneg: rnonneg(x),
rless: x < y,
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
so_lambda: λ2x.t[x],
locally-non-constant-rational: locally-non-constant-rational(f;a;b;c),
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
intermediate-value-lemma',
req_witness,
sq_stable__rleq,
rleq_weakening_rless,
rleq_wf,
r-ap_wf,
rccint_wf,
rleq_weakening_equal,
rleq-int-fractions2,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
istype-false,
rless-int-fractions3,
rdiv_wf,
rless-int,
rless_wf,
int-to-real_wf,
i-member_wf,
rsub_wf,
rmul_wf,
locally-non-constant_wf,
req_wf,
rfun_wf,
real_wf,
sq_stable__and,
sq_stable__i-member,
le_witness_for_triv,
sq_stable__rless,
rmul_preserves_rless,
radd_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
rinv_wf2,
itermAdd_wf,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
nequal_wf,
rless-implies-rless,
req-iff-rsub-is-0,
rless_functionality,
req_transitivity,
radd_functionality,
rmul-rinv3,
int-rinv-cancel,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
locally-non-constant-via-rational,
function-is-continuous,
member_rccint_lemma,
rless_transitivity2,
rless_transitivity1,
nat_plus_wf,
set-value-type,
equal_wf,
less_than_wf,
int-value-type,
set_subtype_base,
int-rdiv_wf,
nat_plus_inc_int_nzero,
rneq_wf,
rneq-int,
rmul_preserves_rleq,
minus-one-mul-top,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
rleq_functionality,
req_weakening
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation_alt,
sqequalRule,
lambdaEquality_alt,
isectElimination,
applyEquality,
because_Cache,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
independent_isectElimination,
setElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
universeIsType,
dependent_pairFormation_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
isect_memberEquality_alt,
voidElimination,
productElimination,
inrFormation_alt,
setIsType,
functionIsType,
equalityTransitivity,
equalitySymmetry,
productEquality,
closedConclusion,
instantiate,
cumulativity,
intEquality,
equalityIstype,
sqequalBase,
int_eqEquality,
cutEval,
promote_hyp,
applyLambdaEquality,
dependent_set_memberFormation_alt,
minusEquality
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a < b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
\mforall{}c:\mBbbR{}
((f(a) \mleq{} c)
{}\mRightarrow{} (c \mleq{} f(b))
{}\mRightarrow{} locally-non-constant(f;a;b;c)
{}\mRightarrow{} (\mexists{}x:\mBbbR{} [((x \mmember{} [a, b]) \mwedge{} (f(x) = c))]))
supposing \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} (f[x] = f[y]))
Date html generated:
2019_10_30-AM-07_49_50
Last ObjectModification:
2019_01_31-AM-11_38_02
Theory : reals
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