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Theorem ioombl1lem4 23542
Description: Lemma for ioombl1 23543. (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 3981 . . . . 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 23467 . . . 4 (((𝐸𝐵) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐵)) ∈ ℝ)
62, 3, 4, 5syl3anc 1476 . . 3 (𝜑 → (vol*‘(𝐸𝐵)) ∈ ℝ)
7 difss 3888 . . . . 5 (𝐸𝐵) ⊆ 𝐸
87a1i 11 . . . 4 (𝜑 → (𝐸𝐵) ⊆ 𝐸)
9 ovolsscl 23467 . . . 4 (((𝐸𝐵) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐵)) ∈ ℝ)
108, 3, 4, 9syl3anc 1476 . . 3 (𝜑 → (vol*‘(𝐸𝐵)) ∈ ℝ)
116, 10readdcld 10269 . 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 23540 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
2614rpred 12068 . . 3 (𝜑𝐶 ∈ ℝ)
274, 26readdcld 10269 . 2 (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
2812, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24ioombl1lem1 23539 . . . . . . . . 9 (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
2928simpld 482 . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
30 eqid 2771 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
3130, 16ovolsf 23453 . . . . . . . 8 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
3229, 31syl 17 . . . . . . 7 (𝜑𝑇:ℕ⟶(0[,)+∞))
33 frn 6191 . . . . . . 7 (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞))
3432, 33syl 17 . . . . . 6 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
35 rge0ssre 12480 . . . . . 6 (0[,)+∞) ⊆ ℝ
3634, 35syl6ss 3764 . . . . 5 (𝜑 → ran 𝑇 ⊆ ℝ)
37 1nn 11231 . . . . . . . 8 1 ∈ ℕ
38 fdm 6189 . . . . . . . . 9 (𝑇:ℕ⟶(0[,)+∞) → dom 𝑇 = ℕ)
3932, 38syl 17 . . . . . . . 8 (𝜑 → dom 𝑇 = ℕ)
4037, 39syl5eleqr 2857 . . . . . . 7 (𝜑 → 1 ∈ dom 𝑇)
41 ne0i 4069 . . . . . . 7 (1 ∈ dom 𝑇 → dom 𝑇 ≠ ∅)
4240, 41syl 17 . . . . . 6 (𝜑 → dom 𝑇 ≠ ∅)
43 dm0rn0 5478 . . . . . . 7 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
4443necon3bii 2995 . . . . . 6 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
4542, 44sylib 208 . . . . 5 (𝜑 → ran 𝑇 ≠ ∅)
4632ffvelrnda 6500 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ (0[,)+∞))
4735, 46sseldi 3750 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ ℝ)
48 eqid 2771 . . . . . . . . . . . . 13 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
4948, 15ovolsf 23453 . . . . . . . . . . . 12 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
5018, 49syl 17 . . . . . . . . . . 11 (𝜑𝑆:ℕ⟶(0[,)+∞))
5150ffvelrnda 6500 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ∈ (0[,)+∞))
5235, 51sseldi 3750 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ∈ ℝ)
5325adantr 466 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
54 simpr 471 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
55 nnuz 11923 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
5654, 55syl6eleq 2860 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
57 simpl 468 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝜑)
58 elfznn 12570 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑗) → 𝑛 ∈ ℕ)
5930ovolfsf 23452 . . . . . . . . . . . . . . 15 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
6029, 59syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
6160ffvelrnda 6500 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞))
6235, 61sseldi 3750 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ)
6357, 58, 62syl2an 583 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ)
6448ovolfsf 23452 . . . . . . . . . . . . . . . 16 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
6518, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
6665ffvelrnda 6500 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞))
67 elrege0 12478 . . . . . . . . . . . . . 14 ((((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)))
6866, 67sylib 208 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)))
6968simpld 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
7057, 58, 69syl2an 583 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
7128simprd 483 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
72 eqid 2771 . . . . . . . . . . . . . . . . . . 19 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
7372ovolfsf 23452 . . . . . . . . . . . . . . . . . 18 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞))
7471, 73syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞))
7574ffvelrnda 6500 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞))
76 elrege0 12478 . . . . . . . . . . . . . . . 16 ((((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
7775, 76sylib 208 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
7877simprd 483 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛))
7977simpld 482 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ)
8062, 79addge01d 10815 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛) ↔ (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))))
8178, 80mpbid 222 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
8212, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24ioombl1lem3 23541 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8381, 82breqtrd 4812 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8457, 58, 83syl2an 583 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8556, 63, 70, 84serle 13056 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗))
8616fveq1i 6331 . . . . . . . . . 10 (𝑇𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗)
8715fveq1i 6331 . . . . . . . . . 10 (𝑆𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗)
8885, 86, 873brtr4g 4820 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ≤ (𝑆𝑗))
89 1zzd 11608 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℤ)
90 eqidd 2772 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
9168simprd 483 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
92 frn 6191 . . . . . . . . . . . . . . . . . . . . 21 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
9350, 92syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
94 icossxr 12456 . . . . . . . . . . . . . . . . . . . 20 (0[,)+∞) ⊆ ℝ*
9593, 94syl6ss 3764 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ran 𝑆 ⊆ ℝ*)
9695adantr 466 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
97 ffn 6183 . . . . . . . . . . . . . . . . . . . 20 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
9850, 97syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑆 Fn ℕ)
99 fnfvelrn 6497 . . . . . . . . . . . . . . . . . . 19 ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
10098, 99sylan 569 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
101 supxrub 12352 . . . . . . . . . . . . . . . . . 18 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑘) ∈ ran 𝑆) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
10296, 100, 101syl2anc 573 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
103102ralrimiva 3115 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
104 breq2 4790 . . . . . . . . . . . . . . . . . 18 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑆𝑘) ≤ 𝑥 ↔ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < )))
105104ralbidv 3135 . . . . . . . . . . . . . . . . 17 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < )))
106105rspcev 3460 . . . . . . . . . . . . . . . 16 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥)
10725, 103, 106syl2anc 573 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥)
10855, 15, 89, 90, 69, 91, 107isumsup2 14778 . . . . . . . . . . . . . 14 (𝜑𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
10993, 35syl6ss 3764 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ⊆ ℝ)
110 fdm 6189 . . . . . . . . . . . . . . . . . . 19 (𝑆:ℕ⟶(0[,)+∞) → dom 𝑆 = ℕ)
11150, 110syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝑆 = ℕ)
11237, 111syl5eleqr 2857 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ dom 𝑆)
113 ne0i 4069 . . . . . . . . . . . . . . . . 17 (1 ∈ dom 𝑆 → dom 𝑆 ≠ ∅)
114112, 113syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝑆 ≠ ∅)
115 dm0rn0 5478 . . . . . . . . . . . . . . . . 17 (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
116115necon3bii 2995 . . . . . . . . . . . . . . . 16 (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
117114, 116sylib 208 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ≠ ∅)
118 breq1 4789 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑆𝑘) → (𝑧𝑥 ↔ (𝑆𝑘) ≤ 𝑥))
119118ralrn 6503 . . . . . . . . . . . . . . . . . 18 (𝑆 Fn ℕ → (∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
12098, 119syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
121120rexbidv 3200 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
122107, 121mpbird 247 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥)
123 supxrre 12355 . . . . . . . . . . . . . . 15 ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥) → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < ))
124109, 117, 122, 123syl3anc 1476 . . . . . . . . . . . . . 14 (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < ))
125108, 124breqtrrd 4814 . . . . . . . . . . . . 13 (𝜑𝑆 ⇝ sup(ran 𝑆, ℝ*, < ))
126125adantr 466 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑆 ⇝ sup(ran 𝑆, ℝ*, < ))
12715, 126syl5eqbrr 4822 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐹)) ⇝ sup(ran 𝑆, ℝ*, < ))
12869adantlr 694 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
12991adantlr 694 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
13055, 54, 127, 128, 129climserle 14594 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
13187, 130syl5eqbr 4821 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
13247, 52, 53, 88, 131letrd 10394 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
133132ralrimiva 3115 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
134 breq2 4790 . . . . . . . . 9 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑇𝑗) ≤ 𝑥 ↔ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < )))
135134ralbidv 3135 . . . . . . . 8 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < )))
136135rspcev 3460 . . . . . . 7 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥)
13725, 133, 136syl2anc 573 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥)
138 ffn 6183 . . . . . . . . 9 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
13932, 138syl 17 . . . . . . . 8 (𝜑𝑇 Fn ℕ)
140 breq1 4789 . . . . . . . . 9 (𝑧 = (𝑇𝑗) → (𝑧𝑥 ↔ (𝑇𝑗) ≤ 𝑥))
141140ralrn 6503 . . . . . . . 8 (𝑇 Fn ℕ → (∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥))
142139, 141syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥))
143142rexbidv 3200 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥))
144137, 143mpbird 247 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥)
145 suprcl 11183 . . . . 5 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥) → sup(ran 𝑇, ℝ, < ) ∈ ℝ)
14636, 45, 144, 145syl3anc 1476 . . . 4 (𝜑 → sup(ran 𝑇, ℝ, < ) ∈ ℝ)
14772, 17ovolsf 23453 . . . . . . . 8 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
14871, 147syl 17 . . . . . . 7 (𝜑𝑈:ℕ⟶(0[,)+∞))
149 frn 6191 . . . . . . 7 (𝑈:ℕ⟶(0[,)+∞) → ran 𝑈 ⊆ (0[,)+∞))
150148, 149syl 17 . . . . . 6 (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
151150, 35syl6ss 3764 . . . . 5 (𝜑 → ran 𝑈 ⊆ ℝ)
152 fdm 6189 . . . . . . . . 9 (𝑈:ℕ⟶(0[,)+∞) → dom 𝑈 = ℕ)
153148, 152syl 17 . . . . . . . 8 (𝜑 → dom 𝑈 = ℕ)
15437, 153syl5eleqr 2857 . . . . . . 7 (𝜑 → 1 ∈ dom 𝑈)
155 ne0i 4069 . . . . . . 7 (1 ∈ dom 𝑈 → dom 𝑈 ≠ ∅)
156154, 155syl 17 . . . . . 6 (𝜑 → dom 𝑈 ≠ ∅)
157 dm0rn0 5478 . . . . . . 7 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
158157necon3bii 2995 . . . . . 6 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
159156, 158sylib 208 . . . . 5 (𝜑 → ran 𝑈 ≠ ∅)
160148ffvelrnda 6500 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ (0[,)+∞))
16135, 160sseldi 3750 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ ℝ)
16257, 58, 79syl2an 583 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ)
163 elrege0 12478 . . . . . . . . . . . . . . . 16 ((((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛)))
16461, 163sylib 208 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛)))
165164simprd 483 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛))
16679, 62addge02d 10816 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛) ↔ (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))))
167165, 166mpbid 222 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
168167, 82breqtrd 4812 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
16957, 58, 168syl2an 583 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
17056, 162, 70, 169serle 13056 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗))
17117fveq1i 6331 . . . . . . . . . 10 (𝑈𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗)
172170, 171, 873brtr4g 4820 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ≤ (𝑆𝑗))
173161, 52, 53, 172, 131letrd 10394 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
174173ralrimiva 3115 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
175 breq2 4790 . . . . . . . . 9 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑈𝑗) ≤ 𝑥 ↔ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < )))
176175ralbidv 3135 . . . . . . . 8 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < )))
177176rspcev 3460 . . . . . . 7 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥)
17825, 174, 177syl2anc 573 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥)
179 ffn 6183 . . . . . . . . 9 (𝑈:ℕ⟶(0[,)+∞) → 𝑈 Fn ℕ)
180148, 179syl 17 . . . . . . . 8 (𝜑𝑈 Fn ℕ)
181 breq1 4789 . . . . . . . . 9 (𝑧 = (𝑈𝑗) → (𝑧𝑥 ↔ (𝑈𝑗) ≤ 𝑥))
182181ralrn 6503 . . . . . . . 8 (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥))
183180, 182syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥))
184183rexbidv 3200 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥))
185178, 184mpbird 247 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥)
186 suprcl 11183 . . . . 5 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥) → sup(ran 𝑈, ℝ, < ) ∈ ℝ)
187151, 159, 185, 186syl3anc 1476 . . . 4 (𝜑 → sup(ran 𝑈, ℝ, < ) ∈ ℝ)
188 ssralv 3815 . . . . . . . . . 10 ((𝐸𝐵) ⊆ 𝐸 → (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
1891, 188ax-mp 5 . . . . . . . . 9 (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
19021breq1i 4793 . . . . . . . . . . . . 13 (𝑃 < 𝑥 ↔ (1st ‘(𝐹𝑛)) < 𝑥)
191 ovolfcl 23447 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
19218, 191sylan 569 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
193192simp1d 1136 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
19421, 193syl5eqel 2854 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
195194adantlr 694 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
1961, 3syl5ss 3763 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸𝐵) ⊆ ℝ)
197196sselda 3752 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐸𝐵)) → 𝑥 ∈ ℝ)
198197adantr 466 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
199 ltle 10326 . . . . . . . . . . . . . . 15 ((𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑃 < 𝑥𝑃𝑥))
200195, 198, 199syl2anc 573 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥𝑃𝑥))
201 simpr 471 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
202 opex 5060 . . . . . . . . . . . . . . . . . . . 20 ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V
20323fvmpt2 6431 . . . . . . . . . . . . . . . . . . . 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⟩)
204201, 202, 203sylancl 574 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
205204fveq2d 6334 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
20613adantr 466 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
207206, 194ifcld 4270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
208192simp2d 1137 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
20922, 208syl5eqel 2854 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
210207, 209ifcld 4270 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
211 op1stg 7325 . . . . . . . . . . . . . . . . . . 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))
212210, 209, 211syl2anc 573 . . . . . . . . . . . . . . . . . 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))
213205, 212eqtrd 2805 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
214213ad2ant2r 741 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (1st ‘(𝐺𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
215210ad2ant2r 741 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
216207ad2ant2r 741 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
217196ad2antrr 705 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (𝐸𝐵) ⊆ ℝ)
218 simplr 752 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥 ∈ (𝐸𝐵))
219217, 218sseldd 3753 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥 ∈ ℝ)
220209ad2ant2r 741 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑄 ∈ ℝ)
221 min1 12218 . . . . . . . . . . . . . . . . . 18 ((if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ ∧ 𝑄 ∈ ℝ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃𝐴, 𝐴, 𝑃))
222216, 220, 221syl2anc 573 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃𝐴, 𝐴, 𝑃))
22313ad2antrr 705 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴 ∈ ℝ)
224 inss2 3982 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸𝐵) ⊆ 𝐵
225224sseli 3748 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐸𝐵) → 𝑥𝐵)
226225ad2antlr 706 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥𝐵)
22713rexrd 10289 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐴 ∈ ℝ*)
228 pnfxr 10292 . . . . . . . . . . . . . . . . . . . . . . . 24 +∞ ∈ ℝ*
229 elioo2 12414 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞)))
230227, 228, 229sylancl 574 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞)))
23112eleq2i 2842 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐵𝑥 ∈ (𝐴(,)+∞))
232 ltpnf 12152 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ ℝ → 𝑥 < +∞)
233232adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝑥 < +∞)
234233pm4.71i 549 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞))
235 df-3an 1073 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞))
236234, 235bitr4i 267 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞))
237230, 231, 2363bitr4g 303 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑥𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥)))
238 simpr 471 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝐴 < 𝑥)
239237, 238syl6bi 243 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑥𝐵𝐴 < 𝑥))
240239ad2antrr 705 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (𝑥𝐵𝐴 < 𝑥))
241226, 240mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴 < 𝑥)
242223, 219, 241ltled 10385 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴𝑥)
243 simprr 756 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑃𝑥)
244 breq1 4789 . . . . . . . . . . . . . . . . . . 19 (𝐴 = if(𝑃𝐴, 𝐴, 𝑃) → (𝐴𝑥 ↔ if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥))
245 breq1 4789 . . . . . . . . . . . . . . . . . . 19 (𝑃 = if(𝑃𝐴, 𝐴, 𝑃) → (𝑃𝑥 ↔ if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥))
246244, 245ifboth 4263 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑥𝑃𝑥) → if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥)
247242, 243, 246syl2anc 573 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥)
248215, 216, 219, 222, 247letrd 10394 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑥)
249214, 248eqbrtrd 4808 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (1st ‘(𝐺𝑛)) ≤ 𝑥)
250249expr 444 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃𝑥 → (1st ‘(𝐺𝑛)) ≤ 𝑥))
251200, 250syld 47 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → (1st ‘(𝐺𝑛)) ≤ 𝑥))
252190, 251syl5bir 233 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < 𝑥 → (1st ‘(𝐺𝑛)) ≤ 𝑥))
25322breq2i 4794 . . . . . . . . . . . . . 14 (𝑥 < 𝑄𝑥 < (2nd ‘(𝐹𝑛)))
254209adantlr 694 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
255 ltle 10326 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑄 ∈ ℝ) → (𝑥 < 𝑄𝑥𝑄))
256198, 254, 255syl2anc 573 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄𝑥𝑄))
257253, 256syl5bir 233 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹𝑛)) → 𝑥𝑄))
258204fveq2d 6334 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
259 op2ndg 7326 . . . . . . . . . . . . . . . . 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)
260210, 209, 259syl2anc 573 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
261258, 260eqtrd 2805 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = 𝑄)
262261adantlr 694 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = 𝑄)
263262breq2d 4798 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 ≤ (2nd ‘(𝐺𝑛)) ↔ 𝑥𝑄))
264257, 263sylibrd 249 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹𝑛)) → 𝑥 ≤ (2nd ‘(𝐺𝑛))))
265252, 264anim12d 596 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
266265reximdva 3165 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐸𝐵)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
267266ralimdva 3111 . . . . . . . . 9 (𝜑 → (∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
268189, 267syl5 34 . . . . . . . 8 (𝜑 → (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
269 ovolfioo 23448 . . . . . . . . 9 ((𝐸 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐸 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
2703, 18, 269syl2anc 573 . . . . . . . 8 (𝜑 → (𝐸 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
271 ovolficc 23449 . . . . . . . . 9 (((𝐸𝐵) ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
272196, 29, 271syl2anc 573 . . . . . . . 8 (𝜑 → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
273268, 270, 2723imtr4d 283 . . . . . . 7 (𝜑 → (𝐸 ran ((,) ∘ 𝐹) → (𝐸𝐵) ⊆ ran ([,] ∘ 𝐺)))
27419, 273mpd 15 . . . . . 6 (𝜑 → (𝐸𝐵) ⊆ ran ([,] ∘ 𝐺))
27516ovollb2 23470 . . . . . 6 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐸𝐵) ⊆ ran ([,] ∘ 𝐺)) → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ*, < ))
27629, 274, 275syl2anc 573 . . . . 5 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ*, < ))
277 supxrre 12355 . . . . . 6 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
27836, 45, 144, 277syl3anc 1476 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
279276, 278breqtrd 4812 . . . 4 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ, < ))
280 ssralv 3815 . . . . . . . . . 10 ((𝐸𝐵) ⊆ 𝐸 → (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
2817, 280ax-mp 5 . . . . . . . . 9 (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
282194adantlr 694 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
2837, 3syl5ss 3763 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸𝐵) ⊆ ℝ)
284283sselda 3752 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐸𝐵)) → 𝑥 ∈ ℝ)
285284adantr 466 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
286282, 285, 199syl2anc 573 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥𝑃𝑥))
287190, 286syl5bir 233 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < 𝑥𝑃𝑥))
288 opex 5060 . . . . . . . . . . . . . . . . . 18 𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V
28924fvmpt2 6431 . . . . . . . . . . . . . . . . . 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)⟩)
290201, 288, 289sylancl 574 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (𝐻𝑛) = ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
291290fveq2d 6334 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
292 op1stg 7325 . . . . . . . . . . . . . . . . 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)
293194, 210, 292syl2anc 573 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
294291, 293eqtrd 2805 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = 𝑃)
295294adantlr 694 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = 𝑃)
296295breq1d 4796 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑃𝑥))
297287, 296sylibrd 249 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < 𝑥 → (1st ‘(𝐻𝑛)) ≤ 𝑥))
298209adantlr 694 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
299285, 298, 255syl2anc 573 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄𝑥𝑄))
300283ad2antrr 705 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐸𝐵) ⊆ ℝ)
301 simplr 752 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ∈ (𝐸𝐵))
302300, 301sseldd 3753 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ∈ ℝ)
30313ad2antrr 705 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝐴 ∈ ℝ)
304194ad2ant2r 741 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑃 ∈ ℝ)
305303, 304ifcld 4270 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
306 eldifn 3884 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐸𝐵) → ¬ 𝑥𝐵)
307306ad2antlr 706 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → ¬ 𝑥𝐵)
308302biantrurd 522 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐴 < 𝑥 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥)))
309237ad2antrr 705 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝑥𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥)))
310308, 309bitr4d 271 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐴 < 𝑥𝑥𝐵))
311307, 310mtbird 314 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → ¬ 𝐴 < 𝑥)
312302, 303lenltd 10383 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝑥𝐴 ↔ ¬ 𝐴 < 𝑥))
313311, 312mpbird 247 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥𝐴)
314 max2 12216 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 ≤ if(𝑃𝐴, 𝐴, 𝑃))
315304, 303, 314syl2anc 573 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝐴 ≤ if(𝑃𝐴, 𝐴, 𝑃))
316302, 303, 305, 313, 315letrd 10394 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ if(𝑃𝐴, 𝐴, 𝑃))
317 simprr 756 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥𝑄)
318 breq2 4790 . . . . . . . . . . . . . . . . . 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)))
319 breq2 4790 . . . . . . . . . . . . . . . . . 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)))
320318, 319ifboth 4263 . . . . . . . . . . . . . . . . 17 ((𝑥 ≤ if(𝑃𝐴, 𝐴, 𝑃) ∧ 𝑥𝑄) → 𝑥 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
321316, 317, 320syl2anc 573 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
322290fveq2d 6334 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
323 op2ndg 7326 . . . . . . . . . . . . . . . . . . 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))
324194, 210, 323syl2anc 573 . . . . . . . . . . . . . . . . . 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))
325322, 324eqtrd 2805 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
326325ad2ant2r 741 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (2nd ‘(𝐻𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
327321, 326breqtrrd 4814 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ (2nd ‘(𝐻𝑛)))
328327expr 444 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥𝑄𝑥 ≤ (2nd ‘(𝐻𝑛))))
329299, 328syld 47 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄𝑥 ≤ (2nd ‘(𝐻𝑛))))
330253, 329syl5bir 233 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹𝑛)) → 𝑥 ≤ (2nd ‘(𝐻𝑛))))
331297, 330anim12d 596 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
332331reximdva 3165 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐸𝐵)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
333332ralimdva 3111 . . . . . . . . 9 (𝜑 → (∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
334281, 333syl5 34 . . . . . . . 8 (𝜑 → (∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
335 ovolficc 23449 . . . . . . . . 9 (((𝐸𝐵) ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐻) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
336283, 71, 335syl2anc 573 . . . . . . . 8 (𝜑 → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐻) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
337334, 270, 3363imtr4d 283 . . . . . . 7 (𝜑 → (𝐸 ran ((,) ∘ 𝐹) → (𝐸𝐵) ⊆ ran ([,] ∘ 𝐻)))
33819, 337mpd 15 . . . . . 6 (𝜑 → (𝐸𝐵) ⊆ ran ([,] ∘ 𝐻))
33917ovollb2 23470 . . . . . 6 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐸𝐵) ⊆ ran ([,] ∘ 𝐻)) → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
34071, 338, 339syl2anc 573 . . . . 5 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
341 supxrre 12355 . . . . . 6 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥) → sup(ran 𝑈, ℝ*, < ) = sup(ran 𝑈, ℝ, < ))
342151, 159, 185, 341syl3anc 1476 . . . . 5 (𝜑 → sup(ran 𝑈, ℝ*, < ) = sup(ran 𝑈, ℝ, < ))
343340, 342breqtrd 4812 . . . 4 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ, < ))
3446, 10, 146, 187, 279, 343le2addd 10846 . . 3 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )))
345 eqidd 2772 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐺)‘𝑛))
34655, 16, 89, 345, 62, 165, 137isumsup2 14778 . . . . 5 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
347 seqex 13003 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐹)) ∈ V
34815, 347eqeltri 2846 . . . . . 6 𝑆 ∈ V
349348a1i 11 . . . . 5 (𝜑𝑆 ∈ V)
350 eqidd 2772 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = (((abs ∘ − ) ∘ 𝐻)‘𝑛))
35155, 17, 89, 350, 79, 78, 178isumsup2 14778 . . . . 5 (𝜑𝑈 ⇝ sup(ran 𝑈, ℝ, < ))
35247recnd 10268 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ ℂ)
353161recnd 10268 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ ℂ)
35462recnd 10268 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℂ)
35557, 58, 354syl2an 583 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℂ)
35679recnd 10268 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℂ)
35757, 58, 356syl2an 583 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℂ)
35882eqcomd 2777 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
35957, 58, 358syl2an 583 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
36056, 355, 357, 359seradd 13043 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗) = ((seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗)))
36186, 171oveq12i 6803 . . . . . 6 ((𝑇𝑗) + (𝑈𝑗)) = ((seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗))
362360, 87, 3613eqtr4g 2830 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) = ((𝑇𝑗) + (𝑈𝑗)))
36355, 89, 346, 349, 351, 352, 353, 362climadd 14563 . . . 4 (𝜑𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )))
364 climuni 14484 . . . 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*, < ))
365363, 125, 364syl2anc 573 . . 3 (𝜑 → (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )) = sup(ran 𝑆, ℝ*, < ))
366344, 365breqtrd 4812 . 2 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ sup(ran 𝑆, ℝ*, < ))
36711, 25, 27, 366, 20letrd 10394 1 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ ((vol*‘𝐸) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  cin 3722  wss 3723  c0 4063  ifcif 4225  cop 4322   cuni 4574   class class class wbr 4786  cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250  ccom 5253   Fn wfn 6024  wf 6025  cfv 6029  (class class class)co 6791  1st c1st 7311  2nd c2nd 7312  supcsup 8500  cc 10134  cr 10135  0cc0 10136  1c1 10137   + caddc 10139  +∞cpnf 10271  *cxr 10273   < clt 10274  cle 10275  cmin 10466  cn 11220  cuz 11886  +crp 12028  (,)cioo 12373  [,)cico 12375  [,]cicc 12376  ...cfz 12526  seqcseq 13001  abscabs 14175  cli 14416  vol*covol 23443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-inf2 8700  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-pre-sup 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-isom 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-sup 8502  df-inf 8503  df-oi 8569  df-card 8963  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-2 11279  df-3 11280  df-n0 11493  df-z 11578  df-uz 11887  df-q 11990  df-rp 12029  df-ioo 12377  df-ico 12379  df-icc 12380  df-fz 12527  df-fzo 12667  df-fl 12794  df-seq 13002  df-exp 13061  df-hash 13315  df-cj 14040  df-re 14041  df-im 14042  df-sqrt 14176  df-abs 14177  df-clim 14420  df-rlim 14421  df-sum 14618  df-ovol 23445
This theorem is referenced by:  ioombl1  23543
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