### Nuprl Lemma : real-continuity3

`∀a,b:ℝ.`
`  ∀f:[a, b] ⟶ℝ`
`    (∀x,y:{x:ℝ| x ∈ [a, b]} .  (f x ≠ f y `` x ≠ y)`
`    `⇐⇒` ∀k:ℕ+. ∃d:{d:ℝ| r0 < d} . ∀x,y:{x:ℝ| x ∈ [a, b]} .  ((|x - y| ≤ d) `` (|(f x) - f y| ≤ (r1/r(k))))) `
`  supposing a ≤ b`

Proof

Definitions occuring in Statement :  rfun: `I ⟶ℝ` rccint: `[l, u]` i-member: `r ∈ I` rdiv: `(x/y)` rneq: `x ≠ y` rleq: `x ≤ y` rless: `x < y` rabs: `|x|` rsub: `x - y` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a` natural_number: `\$n`
Definitions unfolded in proof :  top: `Top` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` decidable: `Dec(P)` sq_exists: `∃x:{A| B[x]}` rless: `x < y` or: `P ∨ Q` guard: `{T}` rneq: `x ≠ y` nat_plus: `ℕ+` so_apply: `x[s]` rfun: `I ⟶ℝ` so_lambda: `λ2x.t[x]` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` prop: `ℙ` real: `ℝ` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` false: `False` implies: `P `` Q` not: `¬A` and: `P ∧ Q` le: `A ≤ B` rnonneg: `rnonneg(x)` rleq: `x ≤ y` uimplies: `b supposing a` member: `t ∈ T` all: `∀x:A. B[x]` continuous: `f[x] continuous for x ∈ I` rccint: `[l, u]` i-approx: `i-approx(I;n)` squash: `↓T` sq_stable: `SqStable(P)` cand: `A c∧ B`
Lemmas referenced :  sq_stable__rless set_wf icompact_wf i-approx_wf real-continuity2 less_than'_wf rsub_wf real_wf nat_plus_wf continuous-rneq rccint_wf i-member_wf all_wf exists_wf rless_wf int-to-real_wf rleq_wf rabs_wf rdiv_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 rfun_wf
Rules used in proof :  computeAll voidEquality isect_memberEquality intEquality int_eqEquality dependent_pairFormation unionElimination inrFormation functionEquality independent_functionElimination because_Cache setEquality dependent_set_memberEquality independent_pairFormation independent_isectElimination equalitySymmetry equalityTransitivity axiomEquality natural_numberEquality minusEquality rename setElimination applyEquality isectElimination voidElimination independent_pairEquality productElimination lambdaEquality sqequalRule introduction isect_memberFormation hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution hypothesis lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution lemma_by_obid cut productEquality imageElimination baseClosed imageMemberEquality dependent_set_memberFormation

Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}
(\mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .    (f  x  \mneq{}  f  y  {}\mRightarrow{}  x  \mneq{}  y)
\mLeftarrow{}{}\mRightarrow{}  \mforall{}k:\mBbbN{}\msupplus{}
\mexists{}d:\{d:\mBbbR{}|  r0  <  d\}
\mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .    ((|x  -  y|  \mleq{}  d)  {}\mRightarrow{}  (|(f  x)  -  f  y|  \mleq{}  (r1/r(k)))))
supposing  a  \mleq{}  b

Date html generated: 2016_05_18-AM-11_12_27
Last ObjectModification: 2016_01_17-AM-00_18_03

Theory : reals

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