Nuprl Lemma : sine-rminus
∀x:ℝ. (sine(-(x)) = -(sine(x)))
Proof
Definitions occuring in Statement :
sine: sine(x),
req: x = y,
rminus: -(x),
real: ℝ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
isOdd: isOdd(n),
true: True,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y)
Lemmas referenced :
sine-is-limit,
rminus_wf,
real_wf,
int-rmul_wf,
fastexp_wf,
int-rdiv_wf,
fact_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
nat_plus_inc_int_nzero,
rnexp_wf,
istype-nat,
ifthenelse_wf,
isOdd_wf,
sine_wf,
bool_wf,
btrue_wf,
series-sum-unique,
series-sum_functionality,
int-rmul_functionality,
int-rdiv_functionality,
rnexp-rminus,
req_weakening,
equal_wf,
squash_wf,
true_wf,
istype-universe,
eq_int_wf,
mod2-add1,
subtype_rel_self,
iff_weakening_equal,
mod2-2n,
series-sum_wf,
int_nzero_wf,
series-sum-linear2,
int-to-real_wf,
rmul_wf,
itermSubtract_wf,
itermMinus_wf,
req-iff-rsub-is-0,
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_minus_lemma,
nat_plus_properties,
rdiv_wf,
rless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rless_wf,
req_functionality,
rmul_functionality,
req_transitivity,
int-rmul-req,
int-rdiv-req,
rinv_wf2,
rminus_functionality,
rinv-mul-as-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
universeIsType,
lambdaEquality_alt,
minusEquality,
natural_numberEquality,
dependent_set_memberEquality_alt,
addEquality,
multiplyEquality,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
applyEquality,
because_Cache,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
intEquality,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
functionIsType,
applyLambdaEquality,
equalityIstype,
closedConclusion,
inrFormation_alt
Latex:
\mforall{}x:\mBbbR{}. (sine(-(x)) = -(sine(x)))
Date html generated:
2019_10_29-AM-10_30_44
Last ObjectModification:
2019_02_01-AM-10_55_54
Theory : reals
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