Nuprl Lemma : rmul-int
∀[a,b:ℤ]. ((r(a) * r(b)) = r(a * b))
Proof
Definitions occuring in Statement :
req: x = y,
rmul: a * b,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
real: ℝ,
int-to-real: r(n),
reg-seq-mul: reg-seq-mul(x;y),
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
int_nzero: ℤ-o,
absval: |i|,
subtract: n - m
Lemmas referenced :
zero-mul,
mul-distributes-right,
add-commutes,
mul-associates,
mul-commutes,
mul-swap,
minus-one-mul,
nat_wf,
nequal_wf,
div-cancel,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
bdd-diff_weakening,
rmul-bdd-diff-reg-seq-mul,
bdd-diff_functionality,
equal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
nat_plus_properties,
subtract_wf,
absval_wf,
all_wf,
nat_plus_wf,
le_wf,
false_wf,
reg-seq-mul_wf,
real_wf,
req_witness,
int-to-real_wf,
rmul_wf,
req-iff-bdd-diff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
multiplyEquality,
productElimination,
independent_isectElimination,
independent_functionElimination,
intEquality,
sqequalRule,
isect_memberEquality,
because_Cache,
applyEquality,
lambdaEquality,
setElimination,
rename,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
divideEquality,
int_eqEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
computeAll,
instantiate,
unionElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a,b:\mBbbZ{}]. ((r(a) * r(b)) = r(a * b))
Date html generated:
2016_05_18-AM-06_51_58
Last ObjectModification:
2016_01_17-AM-01_47_12
Theory : reals
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