Nuprl Lemma : rlog_functionality
∀[x:{x:ℝ| r0 < x} ]. ∀[y:ℝ]. rlog(x) = rlog(y) supposing x = y
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
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,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
top: Top,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rlog: rlog(x),
rfun: I ⟶ℝ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmin_strict_ub,
int-to-real_wf,
rless-int,
sq_stable__rless,
member_rccint_lemma,
rless_transitivity1,
rmin_wf,
rless_wf,
rleq_wf,
rmax_wf,
req_witness,
rlog_wf,
rleq_weakening,
req_wf,
real_wf,
set_wf,
rdiv_wf,
sq_stable__rleq,
i-member_wf,
rccint_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rdiv_functionality,
req_weakening,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_functionality_endpoints
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
setElimination,
rename,
because_Cache,
productElimination,
independent_functionElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
imageElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
inrFormation,
independent_isectElimination,
productEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
setEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 < x\} ]. \mforall{}[y:\mBbbR{}]. rlog(x) = rlog(y) supposing x = y
Date html generated:
2016_10_26-PM-00_27_16
Last ObjectModification:
2016_09_12-PM-05_44_20
Theory : reals_2
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