Nuprl Lemma : rlog-rmul
∀x,y:{t:ℝ| r0 < t} . (rlog(x * y) = (rlog(x) + rlog(y)))
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rless: x < y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
squash: ↓T,
rfun: I ⟶ℝ,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rge: x ≥ y,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
exists: ∃x:A. B[x],
true: True,
less_than': less_than'(a;b),
less_than: a < b,
rsub: x - y
Lemmas referenced :
sq_stable__rless,
int-to-real_wf,
rmul_wf,
rmul-is-positive,
rless_wf,
sq_stable__req,
rlog_wf,
radd_wf,
member_roiint_lemma,
rsub_wf,
real_wf,
i-member_wf,
roiint_wf,
set_wf,
antiderivatives-differ-by-constant,
iproper-roiint,
rdiv_wf,
derivative-rlog,
derivative-const,
derivative-sub,
sq_stable__i-member,
derivative-id,
derivative-const-mul2,
chain-rule,
rdiv_functionality,
req_functionality,
req_wf,
req_weakening,
monotone-maps-compact,
rmul_functionality_wrt_rleq2,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
all_wf,
rleq_wf,
sq_stable__rleq,
rmul-zero-both,
rless_functionality,
rmul_preserves_rless,
derivative_functionality,
rmul_preserves_req,
rinv_wf2,
uiff_transitivity,
rmul_functionality,
rsub_functionality,
rinv-of-rmul,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv,
rmul-rinv3,
rless-int,
rlog1,
radd-rminus-assoc,
radd-assoc,
req_inversion,
rlog_functionality,
radd_functionality,
rmul-one-both,
rminus_wf,
radd-zero-both,
radd-rminus-both,
radd-ac,
radd_comm,
radd-preserves-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
productElimination,
inlFormation,
independent_pairFormation,
productEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
because_Cache,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
independent_isectElimination,
inrFormation,
equalityTransitivity,
equalitySymmetry,
functionEquality,
comment,
computeAll,
int_eqEquality,
intEquality
Latex:
\mforall{}x,y:\{t:\mBbbR{}| r0 < t\} . (rlog(x * y) = (rlog(x) + rlog(y)))
Date html generated:
2017_10_04-PM-10_26_20
Last ObjectModification:
2017_07_28-AM-08_49_57
Theory : reals_2
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