Nuprl Lemma : cosh-ge-1
∀[x:ℝ]. (r1 ≤ cosh(x))
Proof
Definitions occuring in Statement :
cosh: cosh(x),
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
rsub: x - y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
subtype_rel: A ⊆r B,
real: ℝ,
cosh: cosh(x),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top,
rge: x ≥ y,
rgt: x > y
Lemmas referenced :
rnexp2-nonneg,
sinh_wf,
radd-preserves-rleq,
int-to-real_wf,
rnexp_wf,
rminus_wf,
rleq_functionality,
radd_wf,
false_wf,
le_wf,
cosh_wf,
radd_comm,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
trivial-rsub-rleq,
req_weakening,
cosh2-sinh2,
square-rleq-implies,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
expr_wf,
req_wf,
rexp_wf,
rdiv_wf,
rless-int,
rless_wf,
int-rdiv-req,
rmul_preserves_rleq,
rmul_wf,
rmul-zero-both,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_transitivity,
radd_functionality,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
expr-req,
trivial-rleq-radd,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_weakening_rless,
radd_functionality_wrt_rless1,
rexp-positive,
rnexp-one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
because_Cache,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
lambdaEquality,
independent_pairEquality,
voidElimination,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
addLevel,
instantiate,
cumulativity,
intEquality,
baseClosed,
setEquality,
inrFormation,
imageMemberEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[x:\mBbbR{}]. (r1 \mleq{} cosh(x))
Date html generated:
2017_10_04-PM-10_40_46
Last ObjectModification:
2017_06_21-PM-02_13_20
Theory : reals_2
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