Nuprl Lemma : ln_wf
∀a:{a:ℝ| r0 < a} . (ln(a) ∈ {x:ℝ| x = rlog(a)} )
Proof
Definitions occuring in Statement :
ln: ln(a),
rlog: rlog(x),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
ln: ln(a),
uall: ∀[x:A]. B[x],
prop: ℙ,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
rleq: x ≤ y,
rnonneg: rnonneg(x),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
near-log_wf,
rless_wf,
int-to-real_wf,
nat_plus_properties,
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,
log-from_wf,
int-rdiv_wf,
nat_plus_inc_int_nzero,
rleq_wf,
rdiv_wf,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setElimination,
thin,
rename,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_set_memberEquality_alt,
hypothesisEquality,
hypothesis,
universeIsType,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
inhabitedIsType,
productElimination,
because_Cache,
applyEquality,
equalityIstype,
baseClosed,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
setIsType
Latex:
\mforall{}a:\{a:\mBbbR{}| r0 < a\} . (ln(a) \mmember{} \{x:\mBbbR{}| x = rlog(a)\} )
Date html generated:
2019_10_31-AM-06_10_03
Last ObjectModification:
2019_01_28-AM-10_10_51
Theory : reals_2
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