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Theorem ioombl1lem4 23619
Description: Lemma for ioombl1 23620. (Contributed by Mario Carneiro, 16-Jun-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
ioombl1lem4 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ ((vol*‘𝐸) + 𝐶))
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 ioombl1lem4
Dummy variables 𝑥 𝑗 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3992 . . . . 5 (𝐸𝐵) ⊆ 𝐸
21a1i 11 . . . 4 (𝜑 → (𝐸𝐵) ⊆ 𝐸)
3 ioombl1.e . . . 4 (𝜑𝐸 ⊆ ℝ)
4 ioombl1.v . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
5 ovolsscl 23544 . . . 4 (((𝐸𝐵) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐵)) ∈ ℝ)
62, 3, 4, 5syl3anc 1490 . . 3 (𝜑 → (vol*‘(𝐸𝐵)) ∈ ℝ)
7 difss 3899 . . . . 5 (𝐸𝐵) ⊆ 𝐸
87a1i 11 . . . 4 (𝜑 → (𝐸𝐵) ⊆ 𝐸)
9 ovolsscl 23544 . . . 4 (((𝐸𝐵) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐵)) ∈ ℝ)
108, 3, 4, 9syl3anc 1490 . . 3 (𝜑 → (vol*‘(𝐸𝐵)) ∈ ℝ)
116, 10readdcld 10323 . 2 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ∈ ℝ)
12 ioombl1.b . . 3 𝐵 = (𝐴(,)+∞)
13 ioombl1.a . . 3 (𝜑𝐴 ∈ ℝ)
14 ioombl1.c . . 3 (𝜑𝐶 ∈ ℝ+)
15 ioombl1.s . . 3 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
16 ioombl1.t . . 3 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
17 ioombl1.u . . 3 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
18 ioombl1.f1 . . 3 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19 ioombl1.f2 . . 3 (𝜑𝐸 ran ((,) ∘ 𝐹))
20 ioombl1.f3 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
21 ioombl1.p . . 3 𝑃 = (1st ‘(𝐹𝑛))
22 ioombl1.q . . 3 𝑄 = (2nd ‘(𝐹𝑛))
23 ioombl1.g . . 3 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
24 ioombl1.h . . 3 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
2512, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24ioombl1lem2 23617 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
2614rpred 12070 . . 3 (𝜑𝐶 ∈ ℝ)
274, 26readdcld 10323 . 2 (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
2812, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24ioombl1lem1 23616 . . . . . . . . 9 (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
2928simpld 488 . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
30 eqid 2765 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
3130, 16ovolsf 23530 . . . . . . . 8 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
3229, 31syl 17 . . . . . . 7 (𝜑𝑇:ℕ⟶(0[,)+∞))
3332frnd 6230 . . . . . 6 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
34 rge0ssre 12484 . . . . . 6 (0[,)+∞) ⊆ ℝ
3533, 34syl6ss 3773 . . . . 5 (𝜑 → ran 𝑇 ⊆ ℝ)
36 1nn 11287 . . . . . . . 8 1 ∈ ℕ
3732fdmd 6232 . . . . . . . 8 (𝜑 → dom 𝑇 = ℕ)
3836, 37syl5eleqr 2851 . . . . . . 7 (𝜑 → 1 ∈ dom 𝑇)
3938ne0d 4086 . . . . . 6 (𝜑 → dom 𝑇 ≠ ∅)
40 dm0rn0 5510 . . . . . . 7 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
4140necon3bii 2989 . . . . . 6 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
4239, 41sylib 209 . . . . 5 (𝜑 → ran 𝑇 ≠ ∅)
4332ffvelrnda 6549 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ (0[,)+∞))
4434, 43sseldi 3759 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ ℝ)
45 eqid 2765 . . . . . . . . . . . . 13 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
4645, 15ovolsf 23530 . . . . . . . . . . . 12 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
4718, 46syl 17 . . . . . . . . . . 11 (𝜑𝑆:ℕ⟶(0[,)+∞))
4847ffvelrnda 6549 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ∈ (0[,)+∞))
4934, 48sseldi 3759 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ∈ ℝ)
5025adantr 472 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
51 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
52 nnuz 11923 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
5351, 52syl6eleq 2854 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
54 simpl 474 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝜑)
55 elfznn 12577 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑗) → 𝑛 ∈ ℕ)
5630ovolfsf 23529 . . . . . . . . . . . . . . 15 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
5729, 56syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
5857ffvelrnda 6549 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞))
5934, 58sseldi 3759 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ)
6054, 55, 59syl2an 589 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ)
6145ovolfsf 23529 . . . . . . . . . . . . . . . 16 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
6218, 61syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
6362ffvelrnda 6549 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞))
64 elrege0 12482 . . . . . . . . . . . . . 14 ((((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)))
6563, 64sylib 209 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)))
6665simpld 488 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
6754, 55, 66syl2an 589 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
6828simprd 489 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
69 eqid 2765 . . . . . . . . . . . . . . . . . . 19 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
7069ovolfsf 23529 . . . . . . . . . . . . . . . . . 18 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞))
7168, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞))
7271ffvelrnda 6549 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞))
73 elrege0 12482 . . . . . . . . . . . . . . . 16 ((((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
7472, 73sylib 209 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
7574simprd 489 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛))
7674simpld 488 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ)
7759, 76addge01d 10869 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛) ↔ (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))))
7875, 77mpbid 223 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
7912, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24ioombl1lem3 23618 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8078, 79breqtrd 4835 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8154, 55, 80syl2an 589 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8253, 60, 67, 81serle 13063 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗))
8316fveq1i 6376 . . . . . . . . . 10 (𝑇𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗)
8415fveq1i 6376 . . . . . . . . . 10 (𝑆𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗)
8582, 83, 843brtr4g 4843 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ≤ (𝑆𝑗))
86 1zzd 11655 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℤ)
87 eqidd 2766 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8865simprd 489 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8947frnd 6230 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
90 icossxr 12460 . . . . . . . . . . . . . . . . . . . 20 (0[,)+∞) ⊆ ℝ*
9189, 90syl6ss 3773 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ran 𝑆 ⊆ ℝ*)
9291adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
9347ffnd 6224 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑆 Fn ℕ)
94 fnfvelrn 6546 . . . . . . . . . . . . . . . . . . 19 ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
9593, 94sylan 575 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
96 supxrub 12356 . . . . . . . . . . . . . . . . . 18 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑘) ∈ ran 𝑆) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
9792, 95, 96syl2anc 579 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
9897ralrimiva 3113 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
99 brralrspcev 4869 . . . . . . . . . . . . . . . 16 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥)
10025, 98, 99syl2anc 579 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥)
10152, 15, 86, 87, 66, 88, 100isumsup2 14862 . . . . . . . . . . . . . 14 (𝜑𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
10289, 34syl6ss 3773 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ⊆ ℝ)
10347fdmd 6232 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝑆 = ℕ)
10436, 103syl5eleqr 2851 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ dom 𝑆)
105104ne0d 4086 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝑆 ≠ ∅)
106 dm0rn0 5510 . . . . . . . . . . . . . . . . 17 (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
107106necon3bii 2989 . . . . . . . . . . . . . . . 16 (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
108105, 107sylib 209 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ≠ ∅)
109 breq1 4812 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑆𝑘) → (𝑧𝑥 ↔ (𝑆𝑘) ≤ 𝑥))
110109ralrn 6552 . . . . . . . . . . . . . . . . . 18 (𝑆 Fn ℕ → (∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
11193, 110syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
112111rexbidv 3199 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
113100, 112mpbird 248 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥)
114 supxrre 12359 . . . . . . . . . . . . . . 15 ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥) → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < ))
115102, 108, 113, 114syl3anc 1490 . . . . . . . . . . . . . 14 (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < ))
116101, 115breqtrrd 4837 . . . . . . . . . . . . 13 (𝜑𝑆 ⇝ sup(ran 𝑆, ℝ*, < ))
117116adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑆 ⇝ sup(ran 𝑆, ℝ*, < ))
11815, 117syl5eqbrr 4845 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐹)) ⇝ sup(ran 𝑆, ℝ*, < ))
11966adantlr 706 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
12088adantlr 706 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
12152, 51, 118, 119, 120climserle 14678 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
12284, 121syl5eqbr 4844 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
12344, 49, 50, 85, 122letrd 10448 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
124123ralrimiva 3113 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
125 brralrspcev 4869 . . . . . . 7 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥)
12625, 124, 125syl2anc 579 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥)
12732ffnd 6224 . . . . . . . 8 (𝜑𝑇 Fn ℕ)
128 breq1 4812 . . . . . . . . 9 (𝑧 = (𝑇𝑗) → (𝑧𝑥 ↔ (𝑇𝑗) ≤ 𝑥))
129128ralrn 6552 . . . . . . . 8 (𝑇 Fn ℕ → (∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥))
130127, 129syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥))
131130rexbidv 3199 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥))
132126, 131mpbird 248 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥)
133 suprcl 11237 . . . . 5 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥) → sup(ran 𝑇, ℝ, < ) ∈ ℝ)
13435, 42, 132, 133syl3anc 1490 . . . 4 (𝜑 → sup(ran 𝑇, ℝ, < ) ∈ ℝ)
13569, 17ovolsf 23530 . . . . . . . 8 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
13668, 135syl 17 . . . . . . 7 (𝜑𝑈:ℕ⟶(0[,)+∞))
137136frnd 6230 . . . . . 6 (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
138137, 34syl6ss 3773 . . . . 5 (𝜑 → ran 𝑈 ⊆ ℝ)
139136fdmd 6232 . . . . . . . 8 (𝜑 → dom 𝑈 = ℕ)
14036, 139syl5eleqr 2851 . . . . . . 7 (𝜑 → 1 ∈ dom 𝑈)
141140ne0d 4086 . . . . . 6 (𝜑 → dom 𝑈 ≠ ∅)
142 dm0rn0 5510 . . . . . . 7 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
143142necon3bii 2989 . . . . . 6 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
144141, 143sylib 209 . . . . 5 (𝜑 → ran 𝑈 ≠ ∅)
145136ffvelrnda 6549 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ (0[,)+∞))
14634, 145sseldi 3759 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ ℝ)
14754, 55, 76syl2an 589 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ)
148 elrege0 12482 . . . . . . . . . . . . . . . 16 ((((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛)))
14958, 148sylib 209 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛)))
150149simprd 489 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛))
15176, 59addge02d 10870 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛) ↔ (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))))
152150, 151mpbid 223 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
153152, 79breqtrd 4835 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
15454, 55, 153syl2an 589 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
15553, 147, 67, 154serle 13063 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗))
15617fveq1i 6376 . . . . . . . . . 10 (𝑈𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗)
157155, 156, 843brtr4g 4843 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ≤ (𝑆𝑗))
158146, 49, 50, 157, 122letrd 10448 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
159158ralrimiva 3113 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
160 brralrspcev 4869 . . . . . . 7 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥)
16125, 159, 160syl2anc 579 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥)
162136ffnd 6224 . . . . . . . 8 (𝜑𝑈 Fn ℕ)
163 breq1 4812 . . . . . . . . 9 (𝑧 = (𝑈𝑗) → (𝑧𝑥 ↔ (𝑈𝑗) ≤ 𝑥))
164163ralrn 6552 . . . . . . . 8 (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥))
165162, 164syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥))
166165rexbidv 3199 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥))
167161, 166mpbird 248 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥)
168 suprcl 11237 . . . . 5 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥) → sup(ran 𝑈, ℝ, < ) ∈ ℝ)
169138, 144, 167, 168syl3anc 1490 . . . 4 (𝜑 → sup(ran 𝑈, ℝ, < ) ∈ ℝ)
170 ssralv 3826 . . . . . . . . . 10 ((𝐸𝐵) ⊆ 𝐸 → (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
1711, 170ax-mp 5 . . . . . . . . 9 (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
17221breq1i 4816 . . . . . . . . . . . . 13 (𝑃 < 𝑥 ↔ (1st ‘(𝐹𝑛)) < 𝑥)
173 ovolfcl 23524 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
17418, 173sylan 575 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
175174simp1d 1172 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
17621, 175syl5eqel 2848 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
177176adantlr 706 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
1781, 3syl5ss 3772 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸𝐵) ⊆ ℝ)
179178sselda 3761 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐸𝐵)) → 𝑥 ∈ ℝ)
180179adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
181 ltle 10380 . . . . . . . . . . . . . . 15 ((𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑃 < 𝑥𝑃𝑥))
182177, 180, 181syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥𝑃𝑥))
183 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
184 opex 5088 . . . . . . . . . . . . . . . . . . . 20 ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V
18523fvmpt2 6480 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ ∧ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V) → (𝐺𝑛) = ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
186183, 184, 185sylancl 580 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
187186fveq2d 6379 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
18813adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
189188, 176ifcld 4288 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
190174simp2d 1173 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
19122, 190syl5eqel 2848 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
192189, 191ifcld 4288 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
193 op1stg 7378 . . . . . . . . . . . . . . . . . . 19 ((if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
194192, 191, 193syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
195187, 194eqtrd 2799 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
196195ad2ant2r 753 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (1st ‘(𝐺𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
197192ad2ant2r 753 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
198189ad2ant2r 753 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
199178ad2antrr 717 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (𝐸𝐵) ⊆ ℝ)
200 simplr 785 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥 ∈ (𝐸𝐵))
201199, 200sseldd 3762 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥 ∈ ℝ)
202191ad2ant2r 753 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑄 ∈ ℝ)
203 min1 12222 . . . . . . . . . . . . . . . . . 18 ((if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ ∧ 𝑄 ∈ ℝ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃𝐴, 𝐴, 𝑃))
204198, 202, 203syl2anc 579 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃𝐴, 𝐴, 𝑃))
20513ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴 ∈ ℝ)
206 inss2 3993 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸𝐵) ⊆ 𝐵
207206sseli 3757 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐸𝐵) → 𝑥𝐵)
208207ad2antlr 718 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥𝐵)
20913rexrd 10343 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐴 ∈ ℝ*)
210 pnfxr 10346 . . . . . . . . . . . . . . . . . . . . . . . 24 +∞ ∈ ℝ*
211 elioo2 12418 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞)))
212209, 210, 211sylancl 580 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞)))
21312eleq2i 2836 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐵𝑥 ∈ (𝐴(,)+∞))
214 ltpnf 12154 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ ℝ → 𝑥 < +∞)
215214adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝑥 < +∞)
216215pm4.71i 555 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞))
217 df-3an 1109 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞))
218216, 217bitr4i 269 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞))
219212, 213, 2183bitr4g 305 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑥𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥)))
220 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝐴 < 𝑥)
221219, 220syl6bi 244 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑥𝐵𝐴 < 𝑥))
222221ad2antrr 717 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (𝑥𝐵𝐴 < 𝑥))
223208, 222mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴 < 𝑥)
224205, 201, 223ltled 10439 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴𝑥)
225 simprr 789 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑃𝑥)
226 breq1 4812 . . . . . . . . . . . . . . . . . . 19 (𝐴 = if(𝑃𝐴, 𝐴, 𝑃) → (𝐴𝑥 ↔ if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥))
227 breq1 4812 . . . . . . . . . . . . . . . . . . 19 (𝑃 = if(𝑃𝐴, 𝐴, 𝑃) → (𝑃𝑥 ↔ if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥))
228226, 227ifboth 4281 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑥𝑃𝑥) → if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥)
229224, 225, 228syl2anc 579 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥)
230197, 198, 201, 204, 229letrd 10448 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑥)
231196, 230eqbrtrd 4831 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (1st ‘(𝐺𝑛)) ≤ 𝑥)
232231expr 448 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃𝑥 → (1st ‘(𝐺𝑛)) ≤ 𝑥))
233182, 232syld 47 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → (1st ‘(𝐺𝑛)) ≤ 𝑥))
234172, 233syl5bir 234 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < 𝑥 → (1st ‘(𝐺𝑛)) ≤ 𝑥))
23522breq2i 4817 . . . . . . . . . . . . . 14 (𝑥 < 𝑄𝑥 < (2nd ‘(𝐹𝑛)))
236191adantlr 706 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
237 ltle 10380 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑄 ∈ ℝ) → (𝑥 < 𝑄𝑥𝑄))
238180, 236, 237syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄𝑥𝑄))
239235, 238syl5bir 234 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹𝑛)) → 𝑥𝑄))
240186fveq2d 6379 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
241 op2ndg 7379 . . . . . . . . . . . . . . . . 17 ((if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
242192, 191, 241syl2anc 579 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
243240, 242eqtrd 2799 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = 𝑄)
244243adantlr 706 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = 𝑄)
245244breq2d 4821 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 ≤ (2nd ‘(𝐺𝑛)) ↔ 𝑥𝑄))
246239, 245sylibrd 250 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹𝑛)) → 𝑥 ≤ (2nd ‘(𝐺𝑛))))
247234, 246anim12d 602 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
248247reximdva 3163 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐸𝐵)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
249248ralimdva 3109 . . . . . . . . 9 (𝜑 → (∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
250171, 249syl5 34 . . . . . . . 8 (𝜑 → (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
251 ovolfioo 23525 . . . . . . . . 9 ((𝐸 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐸 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
2523, 18, 251syl2anc 579 . . . . . . . 8 (𝜑 → (𝐸 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
253 ovolficc 23526 . . . . . . . . 9 (((𝐸𝐵) ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
254178, 29, 253syl2anc 579 . . . . . . . 8 (𝜑 → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
255250, 252, 2543imtr4d 285 . . . . . . 7 (𝜑 → (𝐸 ran ((,) ∘ 𝐹) → (𝐸𝐵) ⊆ ran ([,] ∘ 𝐺)))
25619, 255mpd 15 . . . . . 6 (𝜑 → (𝐸𝐵) ⊆ ran ([,] ∘ 𝐺))
25716ovollb2 23547 . . . . . 6 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐸𝐵) ⊆ ran ([,] ∘ 𝐺)) → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ*, < ))
25829, 256, 257syl2anc 579 . . . . 5 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ*, < ))
259 supxrre 12359 . . . . . 6 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
26035, 42, 132, 259syl3anc 1490 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
261258, 260breqtrd 4835 . . . 4 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ, < ))
262 ssralv 3826 . . . . . . . . . 10 ((𝐸𝐵) ⊆ 𝐸 → (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
2637, 262ax-mp 5 . . . . . . . . 9 (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
264176adantlr 706 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
2657, 3syl5ss 3772 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸𝐵) ⊆ ℝ)
266265sselda 3761 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐸𝐵)) → 𝑥 ∈ ℝ)
267266adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
268264, 267, 181syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥𝑃𝑥))
269172, 268syl5bir 234 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < 𝑥𝑃𝑥))
270 opex 5088 . . . . . . . . . . . . . . . . . 18 𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V
27124fvmpt2 6480 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ∧ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V) → (𝐻𝑛) = ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
272183, 270, 271sylancl 580 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (𝐻𝑛) = ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
273272fveq2d 6379 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
274 op1stg 7378 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℝ ∧ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
275176, 192, 274syl2anc 579 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
276273, 275eqtrd 2799 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = 𝑃)
277276adantlr 706 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = 𝑃)
278277breq1d 4819 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑃𝑥))
279269, 278sylibrd 250 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < 𝑥 → (1st ‘(𝐻𝑛)) ≤ 𝑥))
280191adantlr 706 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
281267, 280, 237syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄𝑥𝑄))
282265ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐸𝐵) ⊆ ℝ)
283 simplr 785 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ∈ (𝐸𝐵))
284282, 283sseldd 3762 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ∈ ℝ)
28513ad2antrr 717 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝐴 ∈ ℝ)
286176ad2ant2r 753 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑃 ∈ ℝ)
287285, 286ifcld 4288 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
288 eldifn 3895 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐸𝐵) → ¬ 𝑥𝐵)
289288ad2antlr 718 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → ¬ 𝑥𝐵)
290284biantrurd 528 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐴 < 𝑥 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥)))
291219ad2antrr 717 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝑥𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥)))
292290, 291bitr4d 273 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐴 < 𝑥𝑥𝐵))
293289, 292mtbird 316 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → ¬ 𝐴 < 𝑥)
294284, 285lenltd 10437 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝑥𝐴 ↔ ¬ 𝐴 < 𝑥))
295293, 294mpbird 248 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥𝐴)
296 max2 12220 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 ≤ if(𝑃𝐴, 𝐴, 𝑃))
297286, 285, 296syl2anc 579 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝐴 ≤ if(𝑃𝐴, 𝐴, 𝑃))
298284, 285, 287, 295, 297letrd 10448 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ if(𝑃𝐴, 𝐴, 𝑃))
299 simprr 789 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥𝑄)
300 breq2 4813 . . . . . . . . . . . . . . . . . 18 (if(𝑃𝐴, 𝐴, 𝑃) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) → (𝑥 ≤ if(𝑃𝐴, 𝐴, 𝑃) ↔ 𝑥 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)))
301 breq2 4813 . . . . . . . . . . . . . . . . . 18 (𝑄 = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) → (𝑥𝑄𝑥 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)))
302300, 301ifboth 4281 . . . . . . . . . . . . . . . . 17 ((𝑥 ≤ if(𝑃𝐴, 𝐴, 𝑃) ∧ 𝑥𝑄) → 𝑥 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
303298, 299, 302syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
304272fveq2d 6379 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
305 op2ndg 7379 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ ℝ ∧ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
306176, 192, 305syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
307304, 306eqtrd 2799 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
308307ad2ant2r 753 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (2nd ‘(𝐻𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
309303, 308breqtrrd 4837 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ (2nd ‘(𝐻𝑛)))
310309expr 448 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥𝑄𝑥 ≤ (2nd ‘(𝐻𝑛))))
311281, 310syld 47 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄𝑥 ≤ (2nd ‘(𝐻𝑛))))
312235, 311syl5bir 234 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹𝑛)) → 𝑥 ≤ (2nd ‘(𝐻𝑛))))
313279, 312anim12d 602 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
314313reximdva 3163 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐸𝐵)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
315314ralimdva 3109 . . . . . . . . 9 (𝜑 → (∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
316263, 315syl5 34 . . . . . . . 8 (𝜑 → (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
317 ovolficc 23526 . . . . . . . . 9 (((𝐸𝐵) ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐻) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
318265, 68, 317syl2anc 579 . . . . . . . 8 (𝜑 → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐻) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
319316, 252, 3183imtr4d 285 . . . . . . 7 (𝜑 → (𝐸 ran ((,) ∘ 𝐹) → (𝐸𝐵) ⊆ ran ([,] ∘ 𝐻)))
32019, 319mpd 15 . . . . . 6 (𝜑 → (𝐸𝐵) ⊆ ran ([,] ∘ 𝐻))
32117ovollb2 23547 . . . . . 6 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐸𝐵) ⊆ ran ([,] ∘ 𝐻)) → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
32268, 320, 321syl2anc 579 . . . . 5 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
323 supxrre 12359 . . . . . 6 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥) → sup(ran 𝑈, ℝ*, < ) = sup(ran 𝑈, ℝ, < ))
324138, 144, 167, 323syl3anc 1490 . . . . 5 (𝜑 → sup(ran 𝑈, ℝ*, < ) = sup(ran 𝑈, ℝ, < ))
325322, 324breqtrd 4835 . . . 4 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ, < ))
3266, 10, 134, 169, 261, 325le2addd 10900 . . 3 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )))
327 eqidd 2766 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐺)‘𝑛))
32852, 16, 86, 327, 59, 150, 126isumsup2 14862 . . . . 5 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
329 seqex 13010 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐹)) ∈ V
33015, 329eqeltri 2840 . . . . . 6 𝑆 ∈ V
331330a1i 11 . . . . 5 (𝜑𝑆 ∈ V)
332 eqidd 2766 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = (((abs ∘ − ) ∘ 𝐻)‘𝑛))
33352, 17, 86, 332, 76, 75, 161isumsup2 14862 . . . . 5 (𝜑𝑈 ⇝ sup(ran 𝑈, ℝ, < ))
33444recnd 10322 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ ℂ)
335146recnd 10322 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ ℂ)
33659recnd 10322 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℂ)
33754, 55, 336syl2an 589 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℂ)
33876recnd 10322 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℂ)
33954, 55, 338syl2an 589 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℂ)
34079eqcomd 2771 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
34154, 55, 340syl2an 589 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
34253, 337, 339, 341seradd 13050 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗) = ((seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗)))
34383, 156oveq12i 6854 . . . . . 6 ((𝑇𝑗) + (𝑈𝑗)) = ((seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗))
344342, 84, 3433eqtr4g 2824 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) = ((𝑇𝑗) + (𝑈𝑗)))
34552, 86, 328, 331, 333, 334, 335, 344climadd 14647 . . . 4 (𝜑𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )))
346 climuni 14568 . . . 4 ((𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )) ∧ 𝑆 ⇝ sup(ran 𝑆, ℝ*, < )) → (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )) = sup(ran 𝑆, ℝ*, < ))
347345, 116, 346syl2anc 579 . . 3 (𝜑 → (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )) = sup(ran 𝑆, ℝ*, < ))
348326, 347breqtrd 4835 . 2 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ sup(ran 𝑆, ℝ*, < ))
34911, 25, 27, 348, 20letrd 10448 1 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ ((vol*‘𝐸) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  cin 3731  wss 3732  c0 4079  ifcif 4243  cop 4340   cuni 4594   class class class wbr 4809  cmpt 4888   × cxp 5275  dom cdm 5277  ran crn 5278  ccom 5281   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  supcsup 8553  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192  +∞cpnf 10325  *cxr 10327   < clt 10328  cle 10329  cmin 10520  cn 11274  cuz 11886  +crp 12028  (,)cioo 12377  [,)cico 12379  [,]cicc 12380  ...cfz 12533  seqcseq 13008  abscabs 14259  cli 14500  vol*covol 23520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-rlim 14505  df-sum 14702  df-ovol 23522
This theorem is referenced by:  ioombl1  23620
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