### Nuprl Lemma : reg-seq-mul_functionality_wrt_bdd-diff

`∀x1:ℝ. ∀[x2,y1:ℕ+ ⟶ ℤ].  ∀y2:ℝ. (bdd-diff(y1;y2) `` bdd-diff(x1;x2) `` bdd-diff(reg-seq-mul(x1;y1);reg-seq-mul(x2;y2)))`

Proof

Definitions occuring in Statement :  reg-seq-mul: `reg-seq-mul(x;y)` real: `ℝ` bdd-diff: `bdd-diff(f;g)` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` implies: `P `` Q` bdd-diff: `bdd-diff(f;g)` exists: `∃x:A. B[x]` reg-seq-mul: `reg-seq-mul(x;y)` member: `t ∈ T` nat: `ℕ` subtype_rel: `A ⊆r B` int_upper: `{i...}` so_lambda: `λ2x.t[x]` real: `ℝ` nat_plus: `ℕ+` so_apply: `x[s]` prop: `ℙ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` uimplies: `b supposing a` guard: `{T}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` nequal: `a ≠ b ∈ T ` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` int_nzero: `ℤ-o` sq_stable: `SqStable(P)` rev_uimplies: `rev_uimplies(P;Q)` subtract: `n - m` sq_type: `SQType(T)` less_than: `a < b`
Lemmas referenced :  canonical-bound_wf int_upper_wf all_wf nat_plus_wf le_wf absval_wf add_nat_wf false_wf multiply_nat_wf subtype_rel_set nat_wf int_upper_subtype_nat nat_properties decidable__le add-is-int-iff multiply-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf itermAdd_wf itermMultiply_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_mul_lemma int_formula_prop_eq_lemma int_formula_prop_wf equal_wf mul_cancel_in_le subtract_wf nat_plus_properties intformless_wf int_formula_prop_less_lemma absval_nat_plus equal-wf-T-base squash_wf true_wf absval_mul iff_weakening_equal less_than'_wf equal-wf-base bdd-diff_wf real_wf nequal_wf rem_bounds_absval less_than_wf set_wf left_mul_subtract_distrib div_rem_sum2 sq_stable__less_than le_functionality le_weakening add_functionality_wrt_le int-triangle-inequality minus-add minus-minus add-associates minus-one-mul add-swap add-commutes subtype_base_sq int_subtype_base decidable__equal_int itermSubtract_wf int_term_value_subtract_lemma add_functionality_wrt_eq multiply_functionality_wrt_le sq_stable__le absval_pos nat_plus_subtype_nat int_upper_properties absval_sym
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation sqequalHypSubstitution productElimination thin sqequalRule dependent_pairFormation dependent_set_memberEquality addEquality natural_numberEquality multiplyEquality setElimination rename cut hypothesisEquality hypothesis introduction extract_by_obid isectElimination applyEquality lambdaEquality setEquality because_Cache independent_pairFormation independent_isectElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_functionElimination unionElimination pointwiseFunctionality promote_hyp baseClosed baseApply closedConclusion int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination divideEquality imageElimination imageMemberEquality universeEquality independent_pairEquality axiomEquality functionExtensionality functionEquality remainderEquality minusEquality instantiate cumulativity

Latex:
\mforall{}x1:\mBbbR{}
\mforall{}[x2,y1:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].
\mforall{}y2:\mBbbR{}.  (bdd-diff(y1;y2)  {}\mRightarrow{}  bdd-diff(x1;x2)  {}\mRightarrow{}  bdd-diff(reg-seq-mul(x1;y1);reg-seq-mul(x2;y2)))

Date html generated: 2017_10_02-PM-07_15_06
Last ObjectModification: 2017_07_28-AM-07_20_19

Theory : reals

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