Nuprl Lemma : log-by-IVT
∀a:{a:ℝ| r0 < a} . ∃x:ℝ. (x = rlog(a))
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
nat: ℕ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
r-ap: f(x),
rge: x ≥ y,
rgt: x > y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
real: ℝ,
rdiv: (x/y),
rmul: a * b,
int-to-real: r(n),
rinv: rinv(x),
mu-ge: mu-ge(f;n),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
absval: |i|,
bfalse: ff,
btrue: tt,
eq_int: (i =z j),
accelerate: accelerate(k;f),
imax: imax(a;b),
reg-seq-inv: reg-seq-inv(x),
reg-seq-adjust: reg-seq-adjust(n;x),
le_int: i ≤z j,
bnot: ¬bb,
reg-seq-mul: reg-seq-mul(x;y),
le: A ≤ B,
rfun-eq: rfun-eq(I;f;g),
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
locally-non-constant: locally-non-constant(f;a;b;c),
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
cand: A c∧ B
Lemmas referenced :
rless-int,
int-to-real_wf,
rdiv_wf,
rless_wf,
real_wf,
set_wf,
sq_stable__rless,
r-archimedean,
IVT-locally-non-constant,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
real_exp_wf,
i-member_wf,
rccint_wf,
req_wf,
member_rccint_lemma,
rless-int-fractions2,
istype-less_than,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
req_functionality,
rexp_wf,
real_exp-req,
rexp_functionality,
req_weakening,
rleq_functionality,
rless_transitivity2,
radd_wf,
rless_transitivity1,
trivial-rleq-radd,
rleq-int,
istype-false,
rleq_functionality_wrt_implies,
rleq_weakening,
req_inversion,
rexp-of-nonneg-stronger,
derivative-implies-strictly-increasing-simple,
derivative_functionality_wrt_subinterval,
riiint_wf,
subinterval-riiint,
derivative-rexp,
derivative_functionality,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
rleq_wf,
rexp-positive,
rless_functionality,
rless-cases,
rneq_wf,
rneq_functionality,
sq_stable__req,
rlog_wf,
uiff_transitivity,
rlog_functionality,
req_transitivity,
rlog-rexp,
less_than_wf,
rless-int-fractions3,
rmul-rdiv-cancel2,
rmul_wf,
rmul_preserves_rless,
rmul_comm,
rmul-int-rdiv,
rminus_wf,
rlog-inv,
rmul-zero-both,
rminus-rminus,
rminus_functionality
Rules used in proof :
cut,
baseClosed,
hypothesisEquality,
imageMemberEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
because_Cache,
dependent_functionElimination,
inrFormation,
independent_isectElimination,
natural_numberEquality,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
setElimination,
rename,
imageElimination,
minusEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
dependent_set_memberEquality_alt,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
setIsType,
lambdaFormation_alt,
closedConclusion,
inrFormation_alt,
dependent_set_memberFormation_alt,
addEquality,
computeAll,
productIsType,
promote_hyp,
inlFormation_alt,
multiplyEquality,
dependent_set_memberEquality,
dependent_pairFormation,
addLevel
Latex:
\mforall{}a:\{a:\mBbbR{}| r0 < a\} . \mexists{}x:\mBbbR{}. (x = rlog(a))
Date html generated:
2019_10_31-AM-06_10_15
Last ObjectModification:
2019_02_04-PM-11_58_53
Theory : reals_2
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