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Theorem ioombl1lem1 23233
Description: Lemma for ioombl1 23237. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
ioombl1.b 𝐵 = (𝐴(,)+∞)
ioombl1.a (𝜑𝐴 ∈ ℝ)
ioombl1.e (𝜑𝐸 ⊆ ℝ)
ioombl1.v (𝜑 → (vol*‘𝐸) ∈ ℝ)
ioombl1.c (𝜑𝐶 ∈ ℝ+)
ioombl1.s 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ioombl1.t 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ioombl1.u 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ioombl1.f1 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ioombl1.f2 (𝜑𝐸 ran ((,) ∘ 𝐹))
ioombl1.f3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
ioombl1.p 𝑃 = (1st ‘(𝐹𝑛))
ioombl1.q 𝑄 = (2nd ‘(𝐹𝑛))
ioombl1.g 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
ioombl1.h 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
Assertion
Ref Expression
ioombl1lem1 (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
Distinct variable groups:   𝐵,𝑛   𝐶,𝑛   𝑛,𝐸   𝑛,𝐹   𝑛,𝐺   𝑛,𝐻   𝜑,𝑛   𝑆,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝑃(𝑛)   𝑄(𝑛)   𝑇(𝑛)   𝑈(𝑛)

Proof of Theorem ioombl1lem1
StepHypRef Expression
1 ioombl1.a . . . . . . . 8 (𝜑𝐴 ∈ ℝ)
21adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
3 ioombl1.p . . . . . . . 8 𝑃 = (1st ‘(𝐹𝑛))
4 ioombl1.f1 . . . . . . . . . 10 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
5 ovolfcl 23142 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
64, 5sylan 488 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
76simp1d 1071 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
83, 7syl5eqel 2702 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
92, 8ifcld 4103 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
10 ioombl1.q . . . . . . 7 𝑄 = (2nd ‘(𝐹𝑛))
116simp2d 1072 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
1210, 11syl5eqel 2702 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
13 min2 11964 . . . . . 6 ((if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ ∧ 𝑄 ∈ ℝ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑄)
149, 12, 13syl2anc 692 . . . . 5 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑄)
15 df-br 4614 . . . . 5 (if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑄 ↔ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ ≤ )
1614, 15sylib 208 . . . 4 ((𝜑𝑛 ∈ ℕ) → ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ ≤ )
179, 12ifcld 4103 . . . . 5 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
18 opelxpi 5108 . . . . 5 ((if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ (ℝ × ℝ))
1917, 12, 18syl2anc 692 . . . 4 ((𝜑𝑛 ∈ ℕ) → ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ (ℝ × ℝ))
2016, 19elind 3776 . . 3 ((𝜑𝑛 ∈ ℕ) → ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
21 ioombl1.g . . 3 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
2220, 21fmptd 6340 . 2 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23 max1 11959 . . . . . . 7 ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝑃 ≤ if(𝑃𝐴, 𝐴, 𝑃))
248, 2, 23syl2anc 692 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑃 ≤ if(𝑃𝐴, 𝐴, 𝑃))
256simp3d 1073 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
2625, 3, 103brtr4g 4647 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑃𝑄)
27 breq2 4617 . . . . . . 7 (if(𝑃𝐴, 𝐴, 𝑃) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) → (𝑃 ≤ if(𝑃𝐴, 𝐴, 𝑃) ↔ 𝑃 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)))
28 breq2 4617 . . . . . . 7 (𝑄 = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) → (𝑃𝑄𝑃 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)))
2927, 28ifboth 4096 . . . . . 6 ((𝑃 ≤ if(𝑃𝐴, 𝐴, 𝑃) ∧ 𝑃𝑄) → 𝑃 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
3024, 26, 29syl2anc 692 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑃 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
31 df-br 4614 . . . . 5 (𝑃 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ↔ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ ≤ )
3230, 31sylib 208 . . . 4 ((𝜑𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ ≤ )
33 opelxpi 5108 . . . . 5 ((𝑃 ∈ ℝ ∧ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ (ℝ × ℝ))
348, 17, 33syl2anc 692 . . . 4 ((𝜑𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ (ℝ × ℝ))
3532, 34elind 3776 . . 3 ((𝜑𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
36 ioombl1.h . . 3 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
3735, 36fmptd 6340 . 2 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3822, 37jca 554 1 (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cin 3554  wss 3555  ifcif 4058  cop 4154   cuni 4402   class class class wbr 4613  cmpt 4673   × cxp 5072  ran crn 5075  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  supcsup 8290  cr 9879  1c1 9881   + caddc 9883  +∞cpnf 10015  *cxr 10017   < clt 10018  cle 10019  cmin 10210  cn 10964  +crp 11776  (,)cioo 12117  seqcseq 12741  abscabs 13908  vol*covol 23138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1st 7113  df-2nd 7114  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024
This theorem is referenced by:  ioombl1lem3  23235  ioombl1lem4  23236
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