Nuprl Lemma : int-rmul_wf
∀[k:ℤ]. ∀[a:ℝ]. (k * a ∈ ℝ)
Proof
Definitions occuring in Statement :
int-rmul: k1 * a,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
real: ℝ,
int-rmul: k1 * a,
has-value: (a)↓,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
bfalse: ff,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
regular-int-seq: k-regular-seq(f),
nat: ℕ,
le: A ≤ B,
int_lower: {...i},
so_lambda: λ2x.t[x],
so_apply: x[s],
absval: |i|,
subtract: n - m
Lemmas referenced :
value-type-has-value,
int-value-type,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
mul_nat_plus,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
nat_plus_wf,
eqff_to_assert,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
regular-int-seq_wf,
real_wf,
mul_cancel_in_le,
absval_wf,
subtract_wf,
absval_nat_plus,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
squash_wf,
true_wf,
absval_mul,
subtype_rel_self,
iff_weakening_equal,
decidable__equal_int,
multiply-is-int-iff,
add-is-int-iff,
itermMultiply_wf,
itermSubtract_wf,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
istype-nat,
mul-swap,
mul-distributes,
equal_wf,
istype-universe,
add_functionality_wrt_eq,
absval_neg,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
mul-commutes,
set_subtype_base,
left_mul_subtract_distrib,
mul_assoc,
absval_pos,
itermAdd_wf,
int_term_value_add_lemma,
minus-one-mul,
mul-associates,
zero-mul,
zero-add,
add-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
isectElimination,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
dependent_set_memberEquality_alt,
closedConclusion,
natural_numberEquality,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
lessCases,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
lambdaEquality_alt,
minusEquality,
applyEquality,
dependent_functionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
universeIsType,
equalityTransitivity,
equalitySymmetry,
equalityIsType4,
baseApply,
promote_hyp,
instantiate,
cumulativity,
equalityIsType1,
axiomEquality,
multiplyEquality,
addEquality,
universeEquality,
hyp_replacement
Latex:
\mforall{}[k:\mBbbZ{}]. \mforall{}[a:\mBbbR{}]. (k * a \mmember{} \mBbbR{})
Date html generated:
2019_10_29-AM-09_32_16
Last ObjectModification:
2018_11_10-PM-01_33_05
Theory : reals
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