Nuprl Lemma : rlog-rexp
∀[x:ℝ]. (rlog(e^x) = x)
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rexp: e^x,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
prop: ℙ,
so_apply: x[s],
top: Top,
true: True,
guard: {T},
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
or: P ∨ Q,
rneq: x ≠ y,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
itermConstant: "const",
rge: x ≥ y,
rat_term_to_real: rat_term_to_real(f;t),
rtermConstant: "const",
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
pi2: snd(t),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b)
Lemmas referenced :
antiderivatives-equal,
riiint_wf,
iproper-riiint,
int-to-real_wf,
i-member_wf,
rlog_wf,
rexp-positive,
rexp_wf,
rless_wf,
derivative-id,
member_riiint_lemma,
istype-void,
istype-true,
req_inversion,
req_witness,
real_wf,
derivative-rlog,
derivative-rexp,
rdiv_functionality,
set_wf,
req_wf,
req_weakening,
rexp_functionality,
req_functionality,
iproper-roiint,
sq_stable__rless,
rdiv_wf,
member_roiint_lemma,
roiint_wf,
chain-rule,
all_wf,
rleq_wf,
req-iff-rsub-is-0,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_term_value_const_lemma,
itermVar_wf,
itermSubtract_wf,
real_term_polynomial,
rleq_weakening,
rleq_weakening_equal,
rexp_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
monotone-maps-compact,
rmul_wf,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
istype-int,
derivative_functionality,
true_wf,
rless-int,
rlog_functionality,
rexp0,
rlog1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
independent_functionElimination,
sqequalRule,
lambdaEquality_alt,
isectElimination,
natural_numberEquality,
setIsType,
inhabitedIsType,
hypothesisEquality,
universeIsType,
setElimination,
rename,
because_Cache,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
productElimination,
lambdaFormation,
imageElimination,
baseClosed,
imageMemberEquality,
inrFormation,
voidEquality,
isect_memberEquality,
setEquality,
lambdaEquality,
functionEquality,
intEquality,
int_eqEquality,
computeAll,
equalitySymmetry,
equalityTransitivity,
inlFormation,
closedConclusion,
inrFormation_alt,
approximateComputation,
independent_pairFormation,
lambdaFormation_alt,
dependent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}[x:\mBbbR{}]. (rlog(e\^{}x) = x)
Date html generated:
2019_10_31-AM-06_08_27
Last ObjectModification:
2019_04_03-AM-00_28_22
Theory : reals_2
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