Nuprl Lemma : rexp-rlog
∀[x:{x:ℝ| r0 < x} ]. (e^rlog(x) = x)
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rexp: e^x,
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
all: ∀x:A. B[x],
top: Top,
rfun: I ⟶ℝ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
rge: x ≥ y,
rsub: x - y,
false: False
Lemmas referenced :
sq_stable__req,
rexp_wf,
rlog_wf,
rless_wf,
int-to-real_wf,
req_witness,
set_wf,
real_wf,
req-iff-not-rneq,
rneq_wf,
derivative-rlog,
ftc-integral,
roiint_wf,
iproper-roiint,
member_roiint_lemma,
rexp-positive,
i-member_wf,
rdiv_wf,
sq_stable__rless,
req_functionality,
rdiv_functionality,
req_weakening,
req_wf,
all_wf,
member_rccint_lemma,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rmin_strict_ub,
rsub_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_wf,
rminus_wf,
rsub_functionality,
rlog-rexp,
radd-rminus-both,
integral-is-Riemann-on-interval,
rleq_weakening_rless,
integral-reverse,
rminus-zero,
rminus_functionality,
Riemann-integral_wf,
rleq_wf,
rlog-integral-non-zero,
rless_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
lambdaEquality,
productElimination,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
inrFormation,
functionEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
unionElimination
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 < x\} ]. (e\^{}rlog(x) = x)
Date html generated:
2016_10_26-PM-00_28_58
Last ObjectModification:
2016_09_12-PM-05_44_38
Theory : reals_2
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