Nuprl Lemma : real_exp_wf
∀[x:ℝ]. (real_exp(x) ∈ {y:ℝ| y = e^x} )
Proof
Definitions occuring in Statement :
real_exp: real_exp(x),
rexp: e^x,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
real_exp: real_exp(x),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
int-rdiv: (a)/k1,
int-to-real: r(n),
and: P ∧ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
real: ℝ,
nat: ℕ,
int_upper: {i...},
so_apply: x[s],
le: A ≤ B,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
has-value: (a)↓,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
absval: |i|,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
rgt: x > y
Lemmas referenced :
rless-case_wf,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
istype-int,
nequal_wf,
int-to-real_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
real_wf,
canonical-bound-property,
rabs_wf,
int-rmul_wf,
canonical-bound_wf,
subtype_rel_set,
int_upper_wf,
nat_plus_wf,
le_wf,
absval_wf,
istype-int_upper,
subtype_rel_sets_simple,
less_than_wf,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
istype-le,
value-type-has-value,
set-value-type,
int-value-type,
rleq_wf,
rless_wf,
squash_wf,
true_wf,
rabs-int,
subtype_rel_self,
iff_weakening_equal,
rless-int,
nat_wf,
set_subtype_base,
absval-non-neg,
absval_pos,
nat_plus_subtype_nat,
nat_plus_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
nat_plus_inc_int_nzero,
rdiv_wf,
rneq_wf,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rleq_weakening_equal,
rleq_functionality,
rabs_functionality,
int-rdiv-req,
req_weakening,
rabs-rdiv,
req_transitivity,
rmul-rinv3,
rinv-mul-as-rdiv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality_wrt_implies,
uiff_transitivity2,
uiff_transitivity,
int-rmul-req,
rabs-rmul,
rexp-small_wf,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
rnexp_wf,
req_wf,
rexp_wf,
req_functionality,
rnexp_functionality,
rexp_functionality,
rnexp-rexp,
rmul_functionality,
rabs-rleq-iff,
rminus_wf,
rleq_weakening_rless,
itermMinus_wf,
minus-one-mul-top,
int-rinv-cancel2,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityIstype,
baseClosed,
sqequalBase,
universeIsType,
hypothesisEquality,
closedConclusion,
because_Cache,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
addEquality,
applyEquality,
inhabitedIsType,
axiomEquality,
productElimination,
functionEquality,
setElimination,
rename,
multiplyEquality,
callbyvalueReduce,
imageElimination,
universeEquality,
int_eqEquality,
inrFormation_alt,
minusEquality
Latex:
\mforall{}[x:\mBbbR{}]. (real\_exp(x) \mmember{} \{y:\mBbbR{}| y = e\^{}x\} )
Date html generated:
2019_10_30-AM-11_41_19
Last ObjectModification:
2019_02_04-PM-00_17_11
Theory : reals_2
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