Theorem ioombl1lem1 25407

Description: Lemma for ioombl1 25411. (Contributed by Mario Carneiro, 18-Aug-2014.)

Hypotheses

Ref	Expression
ioombl1.b	⊢ 𝐵 = (𝐴(,)+∞)
ioombl1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ioombl1.e	⊢ (𝜑 → 𝐸 ⊆ ℝ)
ioombl1.v	⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
ioombl1.c	⊢ (𝜑 → 𝐶 ∈ ℝ+)
ioombl1.s	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ioombl1.t	⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ioombl1.u	⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ioombl1.f1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ioombl1.f2	⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐹))
ioombl1.f3	⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
ioombl1.p	⊢ 𝑃 = (1st ‘(𝐹‘𝑛))
ioombl1.q	⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))
ioombl1.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
ioombl1.h	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 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

Step	Hyp	Ref	Expression
1		ioombl1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
3		ioombl1.p	. . . . . . . 8 ⊢ 𝑃 = (1st ‘(𝐹‘𝑛))
4		ioombl1.f1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
5		ovolfcl 25315	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
6	4, 5	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
7	6	simp1d 1141	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
8	3, 7	eqeltrid 2836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
9	2, 8	ifcld 4574	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ)
10		ioombl1.q	. . . . . . 7 ⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))
11	6	simp2d 1142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
12	10, 11	eqeltrid 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
13		min2 13176	. . . . . 6 ⊢ ((if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ ∧ 𝑄 ∈ ℝ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑄)
14	9, 12, 13	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑄)
15		df-br 5149	. . . . 5 ⊢ (if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑄 ↔ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ ≤ )
16	14, 15	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ ≤ )
17	9, 12	ifcld 4574	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
18	17, 12	opelxpd 5715	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ (ℝ × ℝ))
19	16, 18	elind 4194	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
20		ioombl1.g	. . 3 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
21	19, 20	fmptd 7115	. 2 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22		max1 13171	. . . . . . 7 ⊢ ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝑃 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃))
23	8, 2, 22	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃))
24	6	simp3d 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
25	24, 3, 10	3brtr4g 5182	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ≤ 𝑄)
26		breq2 5152	. . . . . . 7 ⊢ (if(𝑃 ≤ 𝐴, 𝐴, 𝑃) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) → (𝑃 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ↔ 𝑃 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)))
27		breq2 5152	. . . . . . 7 ⊢ (𝑄 = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) → (𝑃 ≤ 𝑄 ↔ 𝑃 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)))
28	26, 27	ifboth 4567	. . . . . 6 ⊢ ((𝑃 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∧ 𝑃 ≤ 𝑄) → 𝑃 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
29	23, 25, 28	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
30		df-br 5149	. . . . 5 ⊢ (𝑃 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ↔ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ ≤ )
31	29, 30	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ ≤ )
32	8, 17	opelxpd 5715	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ (ℝ × ℝ))
33	31, 32	elind 4194	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
34		ioombl1.h	. . 3 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)
35	33, 34	fmptd 7115	. 2 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
36	21, 35	jca 511	1 ⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ∩ cin 3947   ⊆ wss 3948  ifcif 4528  ⟨cop 4634  ∪ cuni 4908   class class class wbr 5148   ↦ cmpt 5231   × cxp 5674  ran crn 5677   ∘ ccom 5680  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412  1^st c1st 7977  2^nd c2nd 7978  supcsup 9441  ℝcr 11115  1c1 11117   + caddc 11119  +∞cpnf 11252  ℝ^*cxr 11254   < clt 11255   ≤ cle 11256   − cmin 11451  ℕcn 12219  ℝ⁺crp 12981  (,)cioo 13331  seqcseq 13973  abscabs 15188  vol*covol 25311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-pre-lttri 11190  ax-pre-lttrn 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7979  df-2nd 7980  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261

This theorem is referenced by:  ioombl1lem3  25409  ioombl1lem4  25410