Nuprl Lemma : rsin-first-req-1
∀x:ℝ. ((r0 ≤ x) ⇒ (rsin(x) = r1) ⇒ (π/2 ≤ x))
Proof
Definitions occuring in Statement :
halfpi: π/2,
rsin: rsin(x),
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
top: Top,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
rleq-iff-not-rless,
halfpi_wf,
rcos-positive-before-half-pi,
member_rcoint_lemma,
rleq_wf,
int-to-real_wf,
rless_wf,
req_wf,
rsin_wf,
real_wf,
rsin-rcos-pythag,
radd_wf,
rnexp_wf,
false_wf,
le_wf,
rcos_wf,
rmul_wf,
radd-preserves-req,
rminus_wf,
req_functionality,
radd_functionality,
rnexp2,
req_transitivity,
rmul_functionality,
req_weakening,
uiff_transitivity,
rmul-int,
uiff_transitivity3,
squash_wf,
true_wf,
rminus-int,
radd-int,
radd-assoc,
radd-zero-both,
rmul-is-positive,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
dependent_functionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
natural_numberEquality,
because_Cache,
multiplyEquality,
minusEquality,
addEquality,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
inlFormation
Latex:
\mforall{}x:\mBbbR{}. ((r0 \mleq{} x) {}\mRightarrow{} (rsin(x) = r1) {}\mRightarrow{} (\mpi{}/2 \mleq{} x))
Date html generated:
2016_10_26-PM-00_24_04
Last ObjectModification:
2016_09_12-PM-05_43_27
Theory : reals_2
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