### 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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