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Theorem ioombl1lem4 23242
Description: Lemma for ioombl1 23243. (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 3813 . . . . 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 23167 . . . 4 (((𝐸𝐵) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐵)) ∈ ℝ)
62, 3, 4, 5syl3anc 1323 . . 3 (𝜑 → (vol*‘(𝐸𝐵)) ∈ ℝ)
7 difss 3717 . . . . 5 (𝐸𝐵) ⊆ 𝐸
87a1i 11 . . . 4 (𝜑 → (𝐸𝐵) ⊆ 𝐸)
9 ovolsscl 23167 . . . 4 (((𝐸𝐵) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐵)) ∈ ℝ)
108, 3, 4, 9syl3anc 1323 . . 3 (𝜑 → (vol*‘(𝐸𝐵)) ∈ ℝ)
116, 10readdcld 10016 . 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 23240 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
2614rpred 11819 . . 3 (𝜑𝐶 ∈ ℝ)
274, 26readdcld 10016 . 2 (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
2812, 13, 3, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24ioombl1lem1 23239 . . . . . . . . 9 (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
2928simpld 475 . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
30 eqid 2621 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
3130, 16ovolsf 23154 . . . . . . . 8 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
3229, 31syl 17 . . . . . . 7 (𝜑𝑇:ℕ⟶(0[,)+∞))
33 frn 6012 . . . . . . 7 (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞))
3432, 33syl 17 . . . . . 6 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
35 rge0ssre 12225 . . . . . 6 (0[,)+∞) ⊆ ℝ
3634, 35syl6ss 3596 . . . . 5 (𝜑 → ran 𝑇 ⊆ ℝ)
37 1nn 10978 . . . . . . . 8 1 ∈ ℕ
38 fdm 6010 . . . . . . . . 9 (𝑇:ℕ⟶(0[,)+∞) → dom 𝑇 = ℕ)
3932, 38syl 17 . . . . . . . 8 (𝜑 → dom 𝑇 = ℕ)
4037, 39syl5eleqr 2705 . . . . . . 7 (𝜑 → 1 ∈ dom 𝑇)
41 ne0i 3899 . . . . . . 7 (1 ∈ dom 𝑇 → dom 𝑇 ≠ ∅)
4240, 41syl 17 . . . . . 6 (𝜑 → dom 𝑇 ≠ ∅)
43 dm0rn0 5304 . . . . . . 7 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
4443necon3bii 2842 . . . . . 6 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
4542, 44sylib 208 . . . . 5 (𝜑 → ran 𝑇 ≠ ∅)
4632ffvelrnda 6317 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ (0[,)+∞))
4735, 46sseldi 3582 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ ℝ)
48 eqid 2621 . . . . . . . . . . . . 13 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
4948, 15ovolsf 23154 . . . . . . . . . . . 12 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
5018, 49syl 17 . . . . . . . . . . 11 (𝜑𝑆:ℕ⟶(0[,)+∞))
5150ffvelrnda 6317 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ∈ (0[,)+∞))
5235, 51sseldi 3582 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ∈ ℝ)
5325adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
54 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
55 nnuz 11670 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
5654, 55syl6eleq 2708 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
57 simpl 473 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝜑)
58 elfznn 12315 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑗) → 𝑛 ∈ ℕ)
5930ovolfsf 23153 . . . . . . . . . . . . . . 15 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
6029, 59syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
6160ffvelrnda 6317 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞))
6235, 61sseldi 3582 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ)
6357, 58, 62syl2an 494 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ)
6448ovolfsf 23153 . . . . . . . . . . . . . . . 16 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
6518, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
6665ffvelrnda 6317 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞))
67 elrege0 12223 . . . . . . . . . . . . . 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 475 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
7057, 58, 69syl2an 494 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
7128simprd 479 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
72 eqid 2621 . . . . . . . . . . . . . . . . . . 19 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
7372ovolfsf 23153 . . . . . . . . . . . . . . . . . 18 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞))
7471, 73syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞))
7574ffvelrnda 6317 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞))
76 elrege0 12223 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐻)‘𝑛))
7977simpld 475 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ)
8062, 79addge01d 10562 . . . . . . . . . . . . . 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 23241 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8381, 82breqtrd 4641 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8457, 58, 83syl2an 494 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
8556, 63, 70, 84serle 12799 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗))
8616fveq1i 6151 . . . . . . . . . 10 (𝑇𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗)
8715fveq1i 6151 . . . . . . . . . 10 (𝑆𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗)
8885, 86, 873brtr4g 4649 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ≤ (𝑆𝑗))
89 1zzd 11355 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℤ)
90 eqidd 2622 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
9168simprd 479 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
92 frn 6012 . . . . . . . . . . . . . . . . . . . . 21 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
9350, 92syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
94 icossxr 12203 . . . . . . . . . . . . . . . . . . . 20 (0[,)+∞) ⊆ ℝ*
9593, 94syl6ss 3596 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ran 𝑆 ⊆ ℝ*)
9695adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
97 ffn 6004 . . . . . . . . . . . . . . . . . . . 20 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
9850, 97syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑆 Fn ℕ)
99 fnfvelrn 6314 . . . . . . . . . . . . . . . . . . 19 ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
10098, 99sylan 488 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
101 supxrub 12100 . . . . . . . . . . . . . . . . . 18 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑘) ∈ ran 𝑆) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
10296, 100, 101syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
103102ralrimiva 2960 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
104 breq2 4619 . . . . . . . . . . . . . . . . . 18 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑆𝑘) ≤ 𝑥 ↔ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < )))
105104ralbidv 2980 . . . . . . . . . . . . . . . . 17 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < )))
106105rspcev 3295 . . . . . . . . . . . . . . . 16 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥)
10725, 103, 106syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥)
10855, 15, 89, 90, 69, 91, 107isumsup2 14506 . . . . . . . . . . . . . 14 (𝜑𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
10993, 35syl6ss 3596 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ⊆ ℝ)
110 fdm 6010 . . . . . . . . . . . . . . . . . . 19 (𝑆:ℕ⟶(0[,)+∞) → dom 𝑆 = ℕ)
11150, 110syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝑆 = ℕ)
11237, 111syl5eleqr 2705 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ dom 𝑆)
113 ne0i 3899 . . . . . . . . . . . . . . . . 17 (1 ∈ dom 𝑆 → dom 𝑆 ≠ ∅)
114112, 113syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝑆 ≠ ∅)
115 dm0rn0 5304 . . . . . . . . . . . . . . . . 17 (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
116115necon3bii 2842 . . . . . . . . . . . . . . . 16 (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
117114, 116sylib 208 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ≠ ∅)
118 breq1 4618 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑆𝑘) → (𝑧𝑥 ↔ (𝑆𝑘) ≤ 𝑥))
119118ralrn 6320 . . . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . . . 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 12103 . . . . . . . . . . . . . . 15 ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥) → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < ))
124109, 117, 122, 123syl3anc 1323 . . . . . . . . . . . . . 14 (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < ))
125108, 124breqtrrd 4643 . . . . . . . . . . . . 13 (𝜑𝑆 ⇝ sup(ran 𝑆, ℝ*, < ))
126125adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑆 ⇝ sup(ran 𝑆, ℝ*, < ))
12715, 126syl5eqbrr 4651 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐹)) ⇝ sup(ran 𝑆, ℝ*, < ))
12869adantlr 750 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ)
12991adantlr 750 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
13055, 54, 127, 128, 129climserle 14330 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
13187, 130syl5eqbr 4650 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
13247, 52, 53, 88, 131letrd 10141 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
133132ralrimiva 2960 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
134 breq2 4619 . . . . . . . . 9 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑇𝑗) ≤ 𝑥 ↔ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < )))
135134ralbidv 2980 . . . . . . . 8 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < )))
136135rspcev 3295 . . . . . . 7 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥)
13725, 133, 136syl2anc 692 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇𝑗) ≤ 𝑥)
138 ffn 6004 . . . . . . . . 9 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
13932, 138syl 17 . . . . . . . 8 (𝜑𝑇 Fn ℕ)
140 breq1 4618 . . . . . . . . 9 (𝑧 = (𝑇𝑗) → (𝑧𝑥 ↔ (𝑇𝑗) ≤ 𝑥))
141140ralrn 6320 . . . . . . . 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 3045 . . . . . 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 10930 . . . . 5 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥) → sup(ran 𝑇, ℝ, < ) ∈ ℝ)
14636, 45, 144, 145syl3anc 1323 . . . 4 (𝜑 → sup(ran 𝑇, ℝ, < ) ∈ ℝ)
14772, 17ovolsf 23154 . . . . . . . 8 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
14871, 147syl 17 . . . . . . 7 (𝜑𝑈:ℕ⟶(0[,)+∞))
149 frn 6012 . . . . . . 7 (𝑈:ℕ⟶(0[,)+∞) → ran 𝑈 ⊆ (0[,)+∞))
150148, 149syl 17 . . . . . 6 (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
151150, 35syl6ss 3596 . . . . 5 (𝜑 → ran 𝑈 ⊆ ℝ)
152 fdm 6010 . . . . . . . . 9 (𝑈:ℕ⟶(0[,)+∞) → dom 𝑈 = ℕ)
153148, 152syl 17 . . . . . . . 8 (𝜑 → dom 𝑈 = ℕ)
15437, 153syl5eleqr 2705 . . . . . . 7 (𝜑 → 1 ∈ dom 𝑈)
155 ne0i 3899 . . . . . . 7 (1 ∈ dom 𝑈 → dom 𝑈 ≠ ∅)
156154, 155syl 17 . . . . . 6 (𝜑 → dom 𝑈 ≠ ∅)
157 dm0rn0 5304 . . . . . . 7 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
158157necon3bii 2842 . . . . . 6 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
159156, 158sylib 208 . . . . 5 (𝜑 → ran 𝑈 ≠ ∅)
160148ffvelrnda 6317 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ (0[,)+∞))
16135, 160sseldi 3582 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ ℝ)
16257, 58, 79syl2an 494 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ)
163 elrege0 12223 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑛))
16679, 62addge02d 10563 . . . . . . . . . . . . . 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 4641 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
16957, 58, 168syl2an 494 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛))
17056, 162, 70, 169serle 12799 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗))
17117fveq1i 6151 . . . . . . . . . 10 (𝑈𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗)
172170, 171, 873brtr4g 4649 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ≤ (𝑆𝑗))
173161, 52, 53, 172, 131letrd 10141 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
174173ralrimiva 2960 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
175 breq2 4619 . . . . . . . . 9 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝑈𝑗) ≤ 𝑥 ↔ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < )))
176175ralbidv 2980 . . . . . . . 8 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < )))
177176rspcev 3295 . . . . . . 7 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥)
17825, 174, 177syl2anc 692 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ 𝑥)
179 ffn 6004 . . . . . . . . 9 (𝑈:ℕ⟶(0[,)+∞) → 𝑈 Fn ℕ)
180148, 179syl 17 . . . . . . . 8 (𝜑𝑈 Fn ℕ)
181 breq1 4618 . . . . . . . . 9 (𝑧 = (𝑈𝑗) → (𝑧𝑥 ↔ (𝑈𝑗) ≤ 𝑥))
182181ralrn 6320 . . . . . . . 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 3045 . . . . . 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 10930 . . . . 5 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥) → sup(ran 𝑈, ℝ, < ) ∈ ℝ)
187151, 159, 185, 186syl3anc 1323 . . . 4 (𝜑 → sup(ran 𝑈, ℝ, < ) ∈ ℝ)
188 ssralv 3647 . . . . . . . . . 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 4622 . . . . . . . . . . . . 13 (𝑃 < 𝑥 ↔ (1st ‘(𝐹𝑛)) < 𝑥)
191 ovolfcl 23148 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
19218, 191sylan 488 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
193192simp1d 1071 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
19421, 193syl5eqel 2702 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
195194adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
1961, 3syl5ss 3595 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸𝐵) ⊆ ℝ)
197196sselda 3584 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐸𝐵)) → 𝑥 ∈ ℝ)
198197adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
199 ltle 10073 . . . . . . . . . . . . . . 15 ((𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑃 < 𝑥𝑃𝑥))
200195, 198, 199syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥𝑃𝑥))
201 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
202 opex 4895 . . . . . . . . . . . . . . . . . . . 20 ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V
20323fvmpt2 6250 . . . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
205204fveq2d 6154 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
20613adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
207206, 194ifcld 4105 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
208192simp2d 1072 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
20922, 208syl5eqel 2702 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
210207, 209ifcld 4105 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
211 op1stg 7128 . . . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
214213ad2ant2r 782 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (1st ‘(𝐺𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
215210ad2ant2r 782 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
216207ad2ant2r 782 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
217196ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (𝐸𝐵) ⊆ ℝ)
218 simplr 791 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥 ∈ (𝐸𝐵))
219217, 218sseldd 3585 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥 ∈ ℝ)
220209ad2ant2r 782 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑄 ∈ ℝ)
221 min1 11966 . . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃𝐴, 𝐴, 𝑃))
22313ad2antrr 761 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴 ∈ ℝ)
224 inss2 3814 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸𝐵) ⊆ 𝐵
225224sseli 3580 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐸𝐵) → 𝑥𝐵)
226225ad2antlr 762 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑥𝐵)
22713rexrd 10036 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐴 ∈ ℝ*)
228 pnfxr 10039 . . . . . . . . . . . . . . . . . . . . . . . 24 +∞ ∈ ℝ*
229 elioo2 12161 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞)))
230227, 228, 229sylancl 693 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥𝑥 < +∞)))
23112eleq2i 2690 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐵𝑥 ∈ (𝐴(,)+∞))
232 ltpnf 11901 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ ℝ → 𝑥 < +∞)
233232adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝑥 < +∞)
234233pm4.71i 663 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞))
235 df-3an 1038 . . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝐴 < 𝑥)
239237, 238syl6bi 243 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑥𝐵𝐴 < 𝑥))
240239ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (𝑥𝐵𝐴 < 𝑥))
241226, 240mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴 < 𝑥)
242223, 219, 241ltled 10132 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝐴𝑥)
243 simprr 795 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → 𝑃𝑥)
244 breq1 4618 . . . . . . . . . . . . . . . . . . 19 (𝐴 = if(𝑃𝐴, 𝐴, 𝑃) → (𝐴𝑥 ↔ if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥))
245 breq1 4618 . . . . . . . . . . . . . . . . . . 19 (𝑃 = if(𝑃𝐴, 𝐴, 𝑃) → (𝑃𝑥 ↔ if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥))
246244, 245ifboth 4098 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑥𝑃𝑥) → if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥)
247242, 243, 246syl2anc 692 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑥)
248215, 216, 219, 222, 247letrd 10141 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑥)
249214, 248eqbrtrd 4637 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃𝑥)) → (1st ‘(𝐺𝑛)) ≤ 𝑥)
250249expr 642 . . . . . . . . . . . . . 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 4623 . . . . . . . . . . . . . 14 (𝑥 < 𝑄𝑥 < (2nd ‘(𝐹𝑛)))
254209adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
255 ltle 10073 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑄 ∈ ℝ) → (𝑥 < 𝑄𝑥𝑄))
256198, 254, 255syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄𝑥𝑄))
257253, 256syl5bir 233 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹𝑛)) → 𝑥𝑄))
258204fveq2d 6154 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
259 op2ndg 7129 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
261258, 260eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = 𝑄)
262261adantlr 750 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = 𝑄)
263262breq2d 4627 . . . . . . . . . . . . 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 585 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
266265reximdva 3011 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐸𝐵)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
267266ralimdva 2956 . . . . . . . . 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 23149 . . . . . . . . 9 ((𝐸 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐸 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
2703, 18, 269syl2anc 692 . . . . . . . 8 (𝜑 → (𝐸 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐸𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
271 ovolficc 23150 . . . . . . . . 9 (((𝐸𝐵) ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
272196, 29, 271syl2anc 692 . . . . . . . 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 23170 . . . . . 6 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐸𝐵) ⊆ ran ([,] ∘ 𝐺)) → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ*, < ))
27629, 274, 275syl2anc 692 . . . . 5 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ*, < ))
277 supxrre 12103 . . . . . 6 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
27836, 45, 144, 277syl3anc 1323 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
279276, 278breqtrd 4641 . . . 4 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑇, ℝ, < ))
280 ssralv 3647 . . . . . . . . . 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 750 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
2837, 3syl5ss 3595 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸𝐵) ⊆ ℝ)
284283sselda 3584 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐸𝐵)) → 𝑥 ∈ ℝ)
285284adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
286282, 285, 199syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥𝑃𝑥))
287190, 286syl5bir 233 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < 𝑥𝑃𝑥))
288 opex 4895 . . . . . . . . . . . . . . . . . 18 𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V
28924fvmpt2 6250 . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (𝐻𝑛) = ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
291290fveq2d 6154 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
292 op1stg 7128 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
294291, 293eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = 𝑃)
295294adantlr 750 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = 𝑃)
296295breq1d 4625 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑃𝑥))
297287, 296sylibrd 249 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) < 𝑥 → (1st ‘(𝐻𝑛)) ≤ 𝑥))
298209adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
299285, 298, 255syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄𝑥𝑄))
300283ad2antrr 761 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐸𝐵) ⊆ ℝ)
301 simplr 791 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ∈ (𝐸𝐵))
302300, 301sseldd 3585 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ∈ ℝ)
30313ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝐴 ∈ ℝ)
304194ad2ant2r 782 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑃 ∈ ℝ)
305303, 304ifcld 4105 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
306 eldifn 3713 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐸𝐵) → ¬ 𝑥𝐵)
307306ad2antlr 762 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → ¬ 𝑥𝐵)
308302biantrurd 529 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐴 < 𝑥 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥)))
309237ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝑥𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥)))
310308, 309bitr4d 271 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝐴 < 𝑥𝑥𝐵))
311307, 310mtbird 315 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → ¬ 𝐴 < 𝑥)
312302, 303lenltd 10130 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (𝑥𝐴 ↔ ¬ 𝐴 < 𝑥))
313311, 312mpbird 247 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥𝐴)
314 max2 11964 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 ≤ if(𝑃𝐴, 𝐴, 𝑃))
315304, 303, 314syl2anc 692 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝐴 ≤ if(𝑃𝐴, 𝐴, 𝑃))
316302, 303, 305, 313, 315letrd 10141 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ if(𝑃𝐴, 𝐴, 𝑃))
317 simprr 795 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥𝑄)
318 breq2 4619 . . . . . . . . . . . . . . . . . 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 4619 . . . . . . . . . . . . . . . . . 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 4098 . . . . . . . . . . . . . . . . 17 ((𝑥 ≤ if(𝑃𝐴, 𝐴, 𝑃) ∧ 𝑥𝑄) → 𝑥 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
321316, 317, 320syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
322290fveq2d 6154 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
323 op2ndg 7129 . . . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
326325ad2ant2r 782 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → (2nd ‘(𝐻𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
327321, 326breqtrrd 4643 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥𝑄)) → 𝑥 ≤ (2nd ‘(𝐻𝑛)))
328327expr 642 . . . . . . . . . . . . . 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 585 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝐸𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
332331reximdva 3011 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐸𝐵)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
333332ralimdva 2956 . . . . . . . . 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 23150 . . . . . . . . 9 (((𝐸𝐵) ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐸𝐵) ⊆ ran ([,] ∘ 𝐻) ↔ ∀𝑥 ∈ (𝐸𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐻𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐻𝑛)))))
336283, 71, 335syl2anc 692 . . . . . . . 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 23170 . . . . . 6 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐸𝐵) ⊆ ran ([,] ∘ 𝐻)) → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
34071, 338, 339syl2anc 692 . . . . 5 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
341 supxrre 12103 . . . . . 6 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑥) → sup(ran 𝑈, ℝ*, < ) = sup(ran 𝑈, ℝ, < ))
342151, 159, 185, 341syl3anc 1323 . . . . 5 (𝜑 → sup(ran 𝑈, ℝ*, < ) = sup(ran 𝑈, ℝ, < ))
343340, 342breqtrd 4641 . . . 4 (𝜑 → (vol*‘(𝐸𝐵)) ≤ sup(ran 𝑈, ℝ, < ))
3446, 10, 146, 187, 279, 343le2addd 10593 . . 3 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )))
345 eqidd 2622 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐺)‘𝑛))
34655, 16, 89, 345, 62, 165, 137isumsup2 14506 . . . . 5 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
347 seqex 12746 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐹)) ∈ V
34815, 347eqeltri 2694 . . . . . 6 𝑆 ∈ V
349348a1i 11 . . . . 5 (𝜑𝑆 ∈ V)
350 eqidd 2622 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = (((abs ∘ − ) ∘ 𝐻)‘𝑛))
35155, 17, 89, 350, 79, 78, 178isumsup2 14506 . . . . 5 (𝜑𝑈 ⇝ sup(ran 𝑈, ℝ, < ))
35247recnd 10015 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) ∈ ℂ)
353161recnd 10015 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ∈ ℂ)
35462recnd 10015 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℂ)
35557, 58, 354syl2an 494 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) ∈ ℂ)
35679recnd 10015 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℂ)
35757, 58, 356syl2an 494 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) ∈ ℂ)
35882eqcomd 2627 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
35957, 58, 358syl2an 494 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))
36056, 355, 357, 359seradd 12786 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑗) = ((seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗)))
36186, 171oveq12i 6619 . . . . . 6 ((𝑇𝑗) + (𝑈𝑗)) = ((seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝑗))
362360, 87, 3613eqtr4g 2680 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑆𝑗) = ((𝑇𝑗) + (𝑈𝑗)))
36355, 89, 346, 349, 351, 352, 353, 362climadd 14299 . . . 4 (𝜑𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )))
364 climuni 14220 . . . 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 692 . . 3 (𝜑 → (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )) = sup(ran 𝑆, ℝ*, < ))
366344, 365breqtrd 4641 . 2 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ sup(ran 𝑆, ℝ*, < ))
36711, 25, 27, 366, 20letrd 10141 1 (𝜑 → ((vol*‘(𝐸𝐵)) + (vol*‘(𝐸𝐵))) ≤ ((vol*‘𝐸) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3553  cin 3555  wss 3556  c0 3893  ifcif 4060  cop 4156   cuni 4404   class class class wbr 4615  cmpt 4675   × cxp 5074  dom cdm 5076  ran crn 5077  ccom 5080   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  1st c1st 7114  2nd c2nd 7115  supcsup 8293  cc 9881  cr 9882  0cc0 9883  1c1 9884   + caddc 9886  +∞cpnf 10018  *cxr 10020   < clt 10021  cle 10022  cmin 10213  cn 10967  cuz 11634  +crp 11779  (,)cioo 12120  [,)cico 12122  [,]cicc 12123  ...cfz 12271  seqcseq 12744  abscabs 13911  cli 14152  vol*covol 23144
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-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  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-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-sup 8295  df-inf 8296  df-oi 8362  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-n0 11240  df-z 11325  df-uz 11635  df-q 11736  df-rp 11780  df-ioo 12124  df-ico 12126  df-icc 12127  df-fz 12272  df-fzo 12410  df-fl 12536  df-seq 12745  df-exp 12804  df-hash 13061  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-clim 14156  df-rlim 14157  df-sum 14354  df-ovol 23146
This theorem is referenced by:  ioombl1  23243
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