Nuprl Lemma : rabs-rmul
∀[x,y:ℝ]. (|x * y| = (|x| * |y|))
Proof
Definitions occuring in Statement :
rabs: |x|,
req: x = y,
rmul: a * b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
real: ℝ,
reg-seq-mul: reg-seq-mul(x;y),
rabs: |x|,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bdd-diff: bdd-diff(f;g),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
ge: i ≥ j ,
so_apply: x[s],
true: True,
squash: ↓T,
guard: {T},
less_than: a < b,
absval: |i|,
subtract: n - m
Lemmas referenced :
req-iff-bdd-diff,
rabs_wf,
rmul_wf,
bdd-diff_functionality,
absval_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
nat_plus_wf,
reg-seq-mul_wf,
rabs_functionality_wrt_bdd-diff,
rmul-bdd-diff-reg-seq-mul,
false_wf,
le_wf,
all_wf,
subtract_wf,
nat_properties,
req_witness,
real_wf,
nat_wf,
squash_wf,
true_wf,
absval_mul,
iff_weakening_equal,
equal_wf,
absval_div_nat,
mul_nat_plus,
less_than_wf,
minus-one-mul,
add-mul-special,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
divideEquality,
multiplyEquality,
natural_numberEquality,
lambdaFormation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
dependent_set_memberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
universeEquality,
functionExtensionality
Latex:
\mforall{}[x,y:\mBbbR{}]. (|x * y| = (|x| * |y|))
Date html generated:
2017_10_03-AM-08_23_07
Last ObjectModification:
2017_07_28-AM-07_22_43
Theory : reals
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