Nuprl Lemma : rmul_wf
∀[a,b:ℝ]. (a * b ∈ ℝ)
Proof
Definitions occuring in Statement :
rmul: a * b,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rmul: a * b,
has-value: (a)↓,
nat_plus: ℕ+,
real: ℝ,
all: ∀x:A. B[x],
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],
top: Top,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q
Lemmas referenced :
real-has-value,
accelerate_wf,
imax_wf,
absval_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
add_nat_plus,
imax_nat,
nat_plus_properties,
add-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
reg-seq-mul_wf2,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
addEquality,
applyEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
int_eqEquality,
independent_pairFormation,
equalityIstype,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[a,b:\mBbbR{}]. (a * b \mmember{} \mBbbR{})
Date html generated:
2019_10_16-PM-03_06_56
Last ObjectModification:
2019_01_31-PM-04_50_20
Theory : reals
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