Nuprl Lemma : rnexp-rexp
∀[x:ℝ]. ∀[N:ℕ]. (e^x^N = e^r(N) * x)
Proof
Definitions occuring in Statement :
rexp: e^x,
rnexp: x^k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
req_witness,
rnexp_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
rexp_wf,
rmul_wf,
int-to-real_wf,
subtract-1-ge-0,
istype-nat,
real_wf,
rnexp_zero_lemma,
itermSubtract_wf,
itermMultiply_wf,
req_weakening,
req_functionality,
rexp_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rexp0,
subtract-add-cancel,
subtract_wf,
itermAdd_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
radd_wf,
rsub_wf,
rnexp-add1,
rmul_functionality,
req_inversion,
rexp-radd,
req_transitivity,
radd_functionality,
rsub-int,
radd-int,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsTypeImplies,
inhabitedIsType,
dependent_set_memberEquality_alt,
unionElimination,
because_Cache,
isectIsTypeImplies,
productElimination,
addEquality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[N:\mBbbN{}]. (e\^{}x\^{}N = e\^{}r(N) * x)
Date html generated:
2019_10_30-AM-11_40_13
Last ObjectModification:
2019_02_04-PM-00_13_38
Theory : reals_2
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