Nuprl Lemma : ln-req-iff
∀[x:{x:ℝ| r0 < x} ]. ∀[y:ℝ]. uiff(ln(x) = y;x = expr(y))
Proof
Definitions occuring in Statement :
expr: expr(x),
ln: ln(a),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
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,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
req_witness,
expr_wf,
real_wf,
req_wf,
rexp_wf,
ln_wf,
rless_wf,
int-to-real_wf,
rlog_wf,
set_wf,
rless_transitivity1,
rleq_weakening,
ln-expr,
req_functionality,
req_weakening,
expr_functionality,
req_inversion,
ln_functionality,
sq_stable__rless,
expr-ln
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setEquality,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
because_Cache,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 < x\} ]. \mforall{}[y:\mBbbR{}]. uiff(ln(x) = y;x = expr(y))
Date html generated:
2017_10_04-PM-10_38_15
Last ObjectModification:
2017_06_24-AM-11_06_12
Theory : reals_2
Home
Index