Nuprl Lemma : IVT-locally-non-constant-open
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
(∀a',b':ℝ. (((a < a') ∧ (a' < b') ∧ (b' < b)) ⇒ (∀c:ℝ. locally-non-constant(f;a';b';c))))
⇒ (∀c:ℝ. ((f(a) < c) ⇒ (c < f(b)) ⇒ (∃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,
rooint: (l, u),
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
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],
subinterval: I ⊆ J ,
member: t ∈ T,
top: Top,
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
i-member: r ∈ I,
rccint: [l, u],
subtype_rel: A ⊆r B,
continuous: f[x] continuous for x ∈ I,
i-approx: i-approx(I;n),
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
less_than': less_than'(a;b),
rge: x ≥ y,
r-ap: f(x),
rdiv: (x/y)
Lemmas referenced :
member_rooint_lemma,
istype-void,
member_rccint_lemma,
rleq_weakening_rless,
rless_wf,
sq_stable__rless,
req_witness,
function-is-continuous,
r-ap_wf,
rccint_wf,
rleq_weakening_equal,
locally-non-constant_wf,
rfun_subtype_3,
i-member_wf,
req_wf,
rfun_wf,
real_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
rccint-icompact,
icompact_wf,
small-reciprocal-real,
rless-implies-rless,
int-to-real_wf,
rsub_wf,
sq_stable__and,
all_wf,
rleq_wf,
rabs_wf,
rdiv_wf,
rless-int,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
sq_stable__all,
sq_stable__rleq,
le_witness_for_triv,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
ravg-between,
rmin_strict_ub,
radd_wf,
ravg_wf,
trivial-rless-radd,
rmin_strict_lb,
rmin_wf,
rabs-difference-bound-rleq,
radd-preserves-rleq,
rminus_wf,
itermAdd_wf,
itermMinus_wf,
rmin_ub,
rleq_functionality,
real_term_value_add_lemma,
rmin_lb,
real_term_value_minus_lemma,
rmul_wf,
itermMultiply_wf,
rmul-nonneg-case1,
rleq-int,
istype-false,
real_term_value_mul_lemma,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
req_weakening,
rinv_wf2,
req_transitivity,
radd_functionality,
rinv-as-rdiv,
radd-preserves-rless,
rless_functionality_wrt_implies,
rless_functionality,
rmax_strict_lb,
trivial-rsub-rless,
rmax_strict_ub,
rmax_wf,
rless_transitivity2,
rleq-rmax,
rmax_lb,
trivial-rleq-radd,
nat_plus_wf,
IVT-locally-non-constant,
rless_transitivity1,
sq_stable__req,
sq_stable__i-member,
rleq_transitivity,
rleq_weakening,
req_inversion,
continuous-implies-functional
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
productElimination,
isectElimination,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
setElimination,
rename,
because_Cache,
productIsType,
universeIsType,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberFormation_alt,
lambdaEquality_alt,
applyEquality,
functionIsTypeImplies,
inhabitedIsType,
promote_hyp,
functionIsType,
setIsType,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
functionEquality,
productEquality,
closedConclusion,
inrFormation_alt,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
inlFormation_alt
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a < b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
(\mforall{}a',b':\mBbbR{}. (((a < a') \mwedge{} (a' < b') \mwedge{} (b' < b)) {}\mRightarrow{} (\mforall{}c:\mBbbR{}. locally-non-constant(f;a';b';c))))
{}\mRightarrow{} (\mforall{}c:\mBbbR{}. ((f(a) < c) {}\mRightarrow{} (c < f(b)) {}\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-09_09_15
Last ObjectModification:
2019_04_02-AM-11_58_05
Theory : reals
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