### Nuprl Lemma : rabs-nonzero-on-compact

`∀a,b:ℝ.`
`  ((a ≤ b)`
`  `` (∀f:[a, b] ⟶ℝ. ∀k:ℕ+.`
`        (f[x] continuous for x ∈ [a, b]`
`        `` (∀x:ℝ. ((x ∈ [a, b]) `` ((r1/r(k)) ≤ |f[x]|)))`
`        `` ((∀x:ℝ. ((x ∈ [a, b]) `` ((r1/r(k)) ≤ f[x]))) ∨ (∀x:ℝ. ((x ∈ [a, b]) `` (f[x] ≤ (r(-1)/r(k)))))))))`

Proof

Definitions occuring in Statement :  continuous: `f[x] continuous for x ∈ I` rfun: `I ⟶ℝ` rccint: `[l, u]` i-member: `r ∈ I` rdiv: `(x/y)` rleq: `x ≤ y` rabs: `|x|` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` minus: `-n` natural_number: `\$n`
Definitions unfolded in proof :  rev_uimplies: `rev_uimplies(P;Q)` sq_stable: `SqStable(P)` rless: `x < y` sq_exists: `∃x:A [B[x]]` r-ap: `f(x)` less_than: `a < b` squash: `↓T` true: `True` le: `A ≤ B` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)` rleq: `x ≤ y` rnonneg: `rnonneg(x)` subtype_rel: `A ⊆r B` rdiv: `(x/y)` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` i-member: `r ∈ I` rccint: `[l, u]` and: `P ∧ Q` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` nat_plus: `ℕ+` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` so_apply: `x[s]` rfun: `I ⟶ℝ` cand: `A c∧ B` so_lambda: `λ2x.t[x]` label: `...\$L... t`
Lemmas referenced :  r-ap_wf rless_irreflexivity rabs_functionality sq_stable__i-member rmin_ub rmax_lb rleq_transitivity rmin_wf rmax_wf mul_nat_plus less_than_wf rmul_preserves_rless rless-int-fractions intermediate-value-theorem rless_transitivity2 rleq_weakening_rless rless_transitivity1 rless_functionality req_transitivity rmul-rinv rmul_reverses_rleq rminus_wf rleq-int false_wf less_than'_wf rsub_wf rmul_wf rinv_wf2 uiff_transitivity rleq_functionality real_term_polynomial itermSubtract_wf itermMinus_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_mul_lemma real_term_value_minus_lemma real_term_value_var_lemma req-iff-rsub-is-0 req_weakening rminus_functionality rinv-as-rdiv rleq_weakening_equal rabs-ub rdiv_wf int-to-real_wf rless-int nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf rless_wf rless-int-fractions2 itermMultiply_wf int_term_value_mul_lemma member_rccint_lemma rleq_wf i-member_wf rccint_wf all_wf real_wf rabs_wf continuous_wf nat_plus_wf rfun_wf rminus-rdiv rmul-one-both rmul_over_rminus rmul-minus rmul_reverses_rleq_iff squash_wf true_wf rneq_wf rminus-int iff_weakening_equal
Rules used in proof :  universeEquality lemma_by_obid promote_hyp imageElimination imageMemberEquality baseClosed addLevel isect_memberFormation independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination sqequalRule independent_pairFormation introduction extract_by_obid isectElimination because_Cache independent_isectElimination natural_numberEquality setElimination rename inrFormation productElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll multiplyEquality applyEquality dependent_set_memberEquality productEquality inlFormation functionEquality minusEquality setEquality

Latex:
\mforall{}a,b:\mBbbR{}.
((a  \mleq{}  b)
{}\mRightarrow{}  (\mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}.  \mforall{}k:\mBbbN{}\msupplus{}.
(f[x]  continuous  for  x  \mmember{}  [a,  b]
{}\mRightarrow{}  (\mforall{}x:\mBbbR{}.  ((x  \mmember{}  [a,  b])  {}\mRightarrow{}  ((r1/r(k))  \mleq{}  |f[x]|)))
{}\mRightarrow{}  ((\mforall{}x:\mBbbR{}.  ((x  \mmember{}  [a,  b])  {}\mRightarrow{}  ((r1/r(k))  \mleq{}  f[x])))
\mvee{}  (\mforall{}x:\mBbbR{}.  ((x  \mmember{}  [a,  b])  {}\mRightarrow{}  (f[x]  \mleq{}  (r(-1)/r(k)))))))))

Date html generated: 2018_05_22-PM-02_46_47
Last ObjectModification: 2018_05_20-PM-02_47_16

Theory : reals

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