### Nuprl Lemma : rmul-rmin

`∀[x,y,z:ℝ].  ((r0 ≤ z) `` ((z * rmin(x;y)) = rmin(z * x;z * y)))`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rmin: `rmin(x;y)` req: `x = y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` uall: `∀[x:A]. B[x]` implies: `P `` Q` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` real: `ℝ` rmin: `rmin(x;y)` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` reg-seq-mul: `reg-seq-mul(x;y)` bdd-diff: `bdd-diff(f;g)` exists: `∃x:A. B[x]` nat: `ℕ` int_upper: `{i...}` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` false: `False` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` guard: `{T}` ge: `i ≥ j ` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` squash: `↓T` absval: `|i|` subtract: `n - m` cand: `A c∧ B` rev_uimplies: `rev_uimplies(P;Q)` sq_stable: `SqStable(P)`
Lemmas referenced :  req-iff-bdd-diff rmul_wf rmin_wf rleq_wf int-to-real_wf req_witness real_wf reg-seq-mul_wf imin_wf nat_plus_wf bdd-diff_functionality rmul-bdd-diff-reg-seq-mul rmin_functionality_wrt_bdd-diff canonical-bound_wf add_nat_wf multiply_nat_wf decidable__le full-omega-unsat intformnot_wf intformle_wf itermConstant_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_wf istype-le subtype_rel_set int_upper_wf nat_wf le_wf absval_wf istype-int_upper upper_subtype_nat istype-false nat_properties add-is-int-iff multiply-is-int-iff intformand_wf itermVar_wf itermAdd_wf itermMultiply_wf intformeq_wf int_formula_prop_and_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_mul_lemma int_formula_prop_eq_lemma false_wf subtract_wf divide_wfa nat_plus_properties intformless_wf int_formula_prop_less_lemma int_subtype_base nequal_wf decidable__equal_int le_int_wf eqtt_to_assert assert_of_le_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot iff_weakening_uiff assert_wf squash_wf true_wf int_nzero_wf imin_unfold istype-nat subtype_rel_self iff_weakening_equal minus-one-mul mul-commutes add-mul-special zero-mul neg-approx-of-nonneg-real mul_preserves_le decidable__lt istype-less_than div_preserves_le le_functionality int-triangle-inequality2 le_weakening sq_stable__le int_upper_properties absval_div_nat absval_mul div-cancel multiply_functionality_wrt_le
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut lambdaFormation_alt extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination independent_isectElimination universeIsType natural_numberEquality sqequalRule lambdaEquality_alt dependent_functionElimination independent_functionElimination functionIsTypeImplies inhabitedIsType isect_memberEquality_alt because_Cache isectIsTypeImplies applyEquality setElimination rename dependent_pairFormation_alt dependent_set_memberEquality_alt addEquality multiplyEquality equalityTransitivity equalitySymmetry unionElimination approximateComputation voidElimination functionEquality independent_pairFormation applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed int_eqEquality equalityIstype functionIsType sqequalBase intEquality equalityElimination instantiate imageElimination imageMemberEquality universeEquality minusEquality cumulativity

Latex:
\mforall{}[x,y,z:\mBbbR{}].    ((r0  \mleq{}  z)  {}\mRightarrow{}  ((z  *  rmin(x;y))  =  rmin(z  *  x;z  *  y)))

Date html generated: 2019_10_29-AM-10_04_12
Last ObjectModification: 2019_04_01-PM-11_20_10

Theory : reals

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