Nuprl Lemma : arctangent_functionality_wrt_rless
∀x,y:ℝ. arctangent(x) < arctangent(y) supposing x < y
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
rless: x < y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
rge: x ≥ y,
guard: {T},
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
rneq: x ≠ y,
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top,
sq_stable: SqStable(P),
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I
Lemmas referenced :
rnexp2-nonneg,
real_wf,
rless_wf,
int-to-real_wf,
radd_wf,
rnexp_wf,
false_wf,
le_wf,
trivial-rless-radd,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
derivative-implies-strictly-increasing,
riiint_wf,
iproper-riiint,
arctangent_wf,
i-member_wf,
rdiv_wf,
derivative-arctangent,
set_wf,
function-is-continuous,
req_functionality,
rdiv_functionality,
req_weakening,
radd_functionality,
rnexp_functionality,
req_witness,
req_wf,
rmul_preserves_rless,
rmul_wf,
rmul-zero-both,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
req-iff-rsub-is-0,
rless_functionality,
req_transitivity,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
sq_stable__rless,
member_riiint_lemma,
true_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isect_memberFormation,
isectElimination,
natural_numberEquality,
dependent_set_memberEquality,
sqequalRule,
independent_pairFormation,
because_Cache,
productElimination,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
setElimination,
rename,
setEquality,
inrFormation,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination
Latex:
\mforall{}x,y:\mBbbR{}. arctangent(x) < arctangent(y) supposing x < y
Date html generated:
2018_05_22-PM-03_02_08
Last ObjectModification:
2017_10_21-PM-11_25_22
Theory : reals_2
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