Theorem ioombl1lem1 23133

Description: Lemma for ioombl1 23137. (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 23042	. . . . . . . . . 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 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
7	6	simp1d 1066	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
8	3, 7	syl5eqel 2692	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
9	2, 8	ifcld 4081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ)
10		ioombl1.q	. . . . . . 7 ⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))
11	6	simp2d 1067	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
12	10, 11	syl5eqel 2692	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
13		min2 11895	. . . . . 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 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑄)
15		df-br 4584	. . . . 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 207	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ ≤ )
17	9, 12	ifcld 4081	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
18		opelxpi 5072	. . . . 5 ⊢ ((if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ (ℝ × ℝ))
19	17, 12, 18	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ (ℝ × ℝ))
20	16, 19	elind 3760	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
21		ioombl1.g	. . 3 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
22	20, 21	fmptd 6292	. 2 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23		max1 11890	. . . . . . 7 ⊢ ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝑃 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃))
24	8, 2, 23	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃))
25	6	simp3d 1068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
26	25, 3, 10	3brtr4g 4617	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ≤ 𝑄)
27		breq2 4587	. . . . . . 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)))
28		breq2 4587	. . . . . . 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)))
29	27, 28	ifboth 4074	. . . . . 6 ⊢ ((𝑃 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∧ 𝑃 ≤ 𝑄) → 𝑃 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
30	24, 26, 29	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
31		df-br 4584	. . . . 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)⟩ ∈ ≤ )
32	30, 31	sylib 207	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ ≤ )
33		opelxpi 5072	. . . . 5 ⊢ ((𝑃 ∈ ℝ ∧ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ (ℝ × ℝ))
34	8, 17, 33	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ (ℝ × ℝ))
35	32, 34	elind 3760	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
36		ioombl1.h	. . 3 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)
37	35, 36	fmptd 6292	. 2 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
38	22, 37	jca 553	1 ⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  ifcif 4036  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  supcsup 8229  ℝcr 9814  1c1 9816   + caddc 9818  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℝ⁺crp 11708  (,)cioo 12046  seqcseq 12663  abscabs 13822  vol*covol 23038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959

This theorem is referenced by:  ioombl1lem3  23135  ioombl1lem4  23136