Nuprl Lemma : rtan_functionality_wrt_rleq
∀[x,y:{x:ℝ| x ∈ (-(π/2), π/2)} ]. rtan(x) ≤ rtan(y) supposing x ≤ y
Proof
Definitions occuring in Statement :
rtan: rtan(x),
halfpi: π/2,
rooint: (l, u),
i-member: r ∈ I,
rleq: x ≤ y,
rminus: -(x),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
real: ℝ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
nat: ℕ,
less_than': less_than'(a;b),
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
increasing-on-interval: f[x] increasing for x ∈ I,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
rcos-positive,
less_than'_wf,
rsub_wf,
rtan_wf,
i-member_wf,
rooint_wf,
rminus_wf,
halfpi_wf,
real_wf,
nat_plus_wf,
rleq_wf,
set_wf,
rnexp-positive,
rcos_wf,
derivative-implies-increasing,
halfpi-interval-proper,
rdiv_wf,
int-to-real_wf,
rnexp_wf,
false_wf,
le_wf,
rless_wf,
derivative-rtan,
function-is-continuous,
req_functionality,
rdiv_functionality,
req_weakening,
rnexp_functionality,
rcos_functionality,
req_wf,
rmul_preserves_rleq,
rmul_wf,
rmul-zero-both,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
rleq-int,
rleq_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
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
isectElimination,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
lambdaFormation,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
inrFormation,
setEquality,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality
Latex:
\mforall{}[x,y:\{x:\mBbbR{}| x \mmember{} (-(\mpi{}/2), \mpi{}/2)\} ]. rtan(x) \mleq{} rtan(y) supposing x \mleq{} y
Date html generated:
2018_05_22-PM-02_59_36
Last ObjectModification:
2017_10_22-PM-00_38_33
Theory : reals_2
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