Nuprl Lemma : sine-cosine-pythag
∀x:ℝ. ((sine(x)^2 + cosine(x)^2) = r1)
Proof
Definitions occuring in Statement :
cosine: cosine(x),
sine: sine(x),
rnexp: x^k1,
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
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,
prop: ℙ,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
iff: P ⇐⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
top: Top,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
exists: ∃x:A. B[x],
true: True,
guard: {T}
Lemmas referenced :
real_wf,
radd_wf,
rnexp_wf,
false_wf,
le_wf,
sine_wf,
cosine_wf,
rmul_wf,
int-to-real_wf,
derivative-is-zero,
riiint_wf,
iproper-riiint,
i-member_wf,
req_functionality,
radd_functionality,
rnexp2,
req_weakening,
rminus_wf,
member_riiint_lemma,
cosine_functionality,
req_wf,
set_wf,
true_wf,
all_wf,
rminus_functionality,
sine_functionality,
derivative-add,
derivative-mul,
derivative-sine,
derivative-cosine,
derivative_functionality,
uiff_transitivity,
req_transitivity,
rmul_functionality,
rminus-as-rmul,
req_inversion,
rmul-distrib2,
rmul-identity1,
radd-int,
rmul_over_rminus,
rmul-assoc,
rmul_comm,
radd-rminus-both,
cosine0,
sine0,
rmul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
setElimination,
rename,
setEquality,
productElimination,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
applyEquality,
minusEquality,
addEquality,
multiplyEquality
Latex:
\mforall{}x:\mBbbR{}. ((sine(x)\^{}2 + cosine(x)\^{}2) = r1)
Date html generated:
2016_10_26-PM-00_13_14
Last ObjectModification:
2016_09_12-PM-05_39_37
Theory : reals_2
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