Nuprl Lemma : continuous-const
∀[I:Interval]. ∀[a:ℝ]. a continuous for x ∈ I
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
interval: Interval,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
continuous: f[x] continuous for x ∈ I,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_apply: x[s],
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
real: ℝ,
sq_stable: SqStable(P),
absval: |i|,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int-to-real_wf,
rless-int,
rleq_wf,
rabs_wf,
rsub_wf,
i-member_wf,
i-approx_wf,
less_than_wf,
rless_wf,
all_wf,
real_wf,
rdiv_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nat_plus_wf,
set_wf,
icompact_wf,
interval_wf,
absval_wf,
rinv_wf2,
rmul_wf,
rleq-int-fractions2,
sq_stable__less_than,
sq_stable__icompact,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
uiff_transitivity2,
rleq_functionality,
rabs_functionality,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_transitivity,
real_term_value_mul_lemma,
rinv-as-rdiv,
squash_wf,
true_wf,
rabs-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
dependent_set_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
productElimination,
independent_functionElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
dependent_set_memberEquality,
setElimination,
rename,
because_Cache,
productEquality,
lambdaEquality,
functionEquality,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality,
minusEquality,
addEquality,
imageElimination,
multiplyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[I:Interval]. \mforall{}[a:\mBbbR{}]. a continuous for x \mmember{} I
Date html generated:
2017_10_03-AM-10_22_08
Last ObjectModification:
2017_07_28-AM-08_06_25
Theory : reals
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