Nuprl Lemma : not-discontinuous
∀[f:ℝ ⟶ ℝ]. ∀[x:ℝ]. (¬discontinuous(f;x)) supposing ∀x,y:ℝ. ((x = y) ⇒ ((f x) = (f y)))
Proof
Definitions occuring in Statement :
discontinuous: discontinuous(f;x),
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
discontinuous: discontinuous(f;x),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
top: Top,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rsub: x - y,
real-fun: real-fun(f;a;b),
guard: {T},
real-cont: real-cont(f;a;b),
cand: A c∧ B,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
rge: x ≥ y,
rgt: x > y,
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
discontinuous_wf,
real_wf,
all_wf,
req_wf,
real-fun-iff-continuous,
rsub_wf,
int-to-real_wf,
radd_wf,
member_rccint_lemma,
subtype_rel_dep_function,
rleq_wf,
subtype_rel_self,
set_wf,
radd-preserves-rless,
rless-int,
rminus_wf,
rless_wf,
rless_functionality,
req_weakening,
radd-int,
radd_functionality,
radd-zero-both,
req_transitivity,
radd-rminus-both,
radd-rminus-assoc,
radd-ac,
radd_comm,
req_inversion,
radd-assoc,
i-member_wf,
rccint_wf,
small-reciprocal-real,
rmin_wf,
rmin_strict_ub,
sq_stable__rless,
radd-preserves-rleq,
rmul_wf,
rleq-int,
false_wf,
uiff_transitivity,
rleq_functionality,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
uiff_transitivity2,
rmul-zero-both,
squash_wf,
true_wf,
rminus-int,
rabs-difference-bound-iff,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
rmin-rleq,
radd_functionality_wrt_rless2,
rabs_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,
rless_transitivity2,
rless_transitivity1,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
functionExtensionality,
applyEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
natural_numberEquality,
independent_isectElimination,
voidEquality,
setEquality,
productEquality,
setElimination,
rename,
independent_pairFormation,
imageMemberEquality,
baseClosed,
addEquality,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
imageElimination,
minusEquality,
promote_hyp,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[f:\mBbbR{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}]. (\mneg{}discontinuous(f;x)) supposing \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} ((f x) = (f y)))
Date html generated:
2016_10_26-AM-09_51_59
Last ObjectModification:
2016_08_12-PM-07_25_27
Theory : reals
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