Nuprl Lemma : intermediate-value-lemma

a:ℝ. ∀b:{b:ℝa < b} . ∀f:[a, b] ⟶ℝ.
  ∀c:{c:ℝ(f(a) ≤ c) ∧ (c ≤ f(b))} 
    ((∃d:{d:ℝ(r0 ≤ d) ∧ (d < r1)} 
       ∀a1:{a1:ℝ(a1 ∈ [a, b]) ∧ (f(a1) ≤ c)} . ∀b1:{b1:ℝ(b1 ∈ [a, b]) ∧ (c ≤ f(b1)) ∧ (a1 ≤ b1)} .
         ∃a2:{a2:ℝ(a2 ∈ [a, b]) ∧ (f(a2) ≤ c)} (∃b2:{b2:ℝ(b2 ∈ [a, b]) ∧ (c ≤ f(b2))}  [((a1 ≤ a2) ∧ (a2 ≤ b2) ∧ (\000Cb2 ≤ b1) ∧ ((b2 a2) ≤ ((b1 a1) d)))]))
     (∃x:ℝ [((x ∈ [a, b]) ∧ (f(x) c))])) 
  supposing ∀x,y:{x:ℝx ∈ [a, b]} .  ((x y)  (f[x] f[y]))


Proof




Definitions occuring in Statement :  r-ap: f(x) rfun: I ⟶ℝ rccint: [l, u] i-member: r ∈ I rleq: x ≤ y rless: x < y rsub: y req: y rmul: b int-to-real: r(n) real: uimplies: supposing a so_apply: x[s] all: x:A. B[x] sq_exists: x:A [B[x]] exists: x:A. B[x] implies:  Q and: P ∧ Q set: {x:A| B[x]}  natural_number: $n
Definitions unfolded in proof :  all: x:A. B[x] uimplies: supposing a member: t ∈ T implies:  Q uall: [x:A]. B[x] so_apply: x[s] rfun: I ⟶ℝ sq_stable: SqStable(P) guard: {T} squash: T superlevelset: superlevelset(I;f;c) sublevelset: sublevelset(I;f;c) top: Top exists: x:A. B[x] and: P ∧ Q cand: c∧ B prop: i-member: r ∈ I rccint: [l, u] sq_exists: x:A [B[x]] subtype_rel: A ⊆B so_lambda: λ2x.t[x] r-ap: f(x) i-closed: i-closed(I) isl: isl(x) outl: outl(x) bnot: ¬bb ifthenelse: if then else fi  btrue: tt bor: p ∨bq bfalse: ff assert: b true: True closed-rset: closed-rset(A) member-closure: y ∈ closure(A) rleq: x ≤ y rnonneg: rnonneg(x) le: A ≤ B
Lemmas referenced :  req_witness sq_stable__rleq rleq_weakening_rless closures-meet-sq-ext sublevelset_wf rccint_wf superlevelset_wf member_rccint_lemma istype-void rleq_weakening_equal rleq_wf r-ap_wf subtype_rel_sets_simple real_wf i-member_wf rsub_wf rmul_wf function-is-continuous req_wf int-to-real_wf rless_wf rfun_wf sublevelset-closed sq_stable__and le_witness_for_triv istype-nat converges-to_wf subtype_rel_self superlevelset-closed rleq_antisymmetry
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt isect_memberFormation_alt cut introduction sqequalRule sqequalHypSubstitution lambdaEquality_alt dependent_functionElimination thin hypothesisEquality extract_by_obid isectElimination applyEquality because_Cache independent_functionElimination hypothesis functionIsTypeImplies inhabitedIsType rename setElimination independent_isectElimination imageMemberEquality baseClosed imageElimination isect_memberEquality_alt voidElimination dependent_pairFormation_alt independent_pairFormation productElimination dependent_set_memberEquality_alt productIsType universeIsType dependent_set_memberFormation_alt setIsType productEquality functionIsType natural_numberEquality promote_hyp equalityTransitivity equalitySymmetry instantiate universeEquality

Latex:
\mforall{}a:\mBbbR{}.  \mforall{}b:\{b:\mBbbR{}|  a  <  b\}  .  \mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}.
    \mforall{}c:\{c:\mBbbR{}|  (f(a)  \mleq{}  c)  \mwedge{}  (c  \mleq{}  f(b))\} 
        ((\mexists{}d:\{d:\mBbbR{}|  (r0  \mleq{}  d)  \mwedge{}  (d  <  r1)\} 
              \mforall{}a1:\{a1:\mBbbR{}|  (a1  \mmember{}  [a,  b])  \mwedge{}  (f(a1)  \mleq{}  c)\}  .  \mforall{}b1:\{b1:\mBbbR{}|  (b1  \mmember{}  [a,  b])  \mwedge{}  (c  \mleq{}  f(b1))  \mwedge{}  (a1  \mleq{}  b1)\}\000C  .
                  \mexists{}a2:\{a2:\mBbbR{}|  (a2  \mmember{}  [a,  b])  \mwedge{}  (f(a2)  \mleq{}  c)\} 
                    (\mexists{}b2:\{b2:\mBbbR{}|  (b2  \mmember{}  [a,  b])  \mwedge{}  (c  \mleq{}  f(b2))\}    [((a1  \mleq{}  a2)
                                                                                            \mwedge{}  (a2  \mleq{}  b2)
                                                                                            \mwedge{}  (b2  \mleq{}  b1)
                                                                                            \mwedge{}  ((b2  -  a2)  \mleq{}  ((b1  -  a1)  *  d)))]))
        {}\mRightarrow{}  (\mexists{}x:\mBbbR{}  [((x  \mmember{}  [a,  b])  \mwedge{}  (f(x)  =  c))])) 
    supposing  \mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .    ((x  =  y)  {}\mRightarrow{}  (f[x]  =  f[y]))



Date html generated: 2019_10_30-AM-07_48_08
Last ObjectModification: 2019_01_08-PM-11_16_45

Theory : reals


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