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Theorem ovoliunlem1 23316
Description: Lemma for ovoliun 23319. (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovoliun.t 𝑇 = seq1( + , 𝐺)
ovoliun.g 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
ovoliun.r (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
ovoliun.b (𝜑𝐵 ∈ ℝ+)
ovoliun.s 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))
ovoliun.u 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ovoliun.h 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))
ovoliun.j (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))
ovoliun.f (𝜑𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
ovoliun.x1 ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
ovoliun.x2 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
ovoliun.k (𝜑𝐾 ∈ ℕ)
ovoliun.l1 (𝜑𝐿 ∈ ℤ)
ovoliun.l2 (𝜑 → ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽𝑤)) ≤ 𝐿)
Assertion
Ref Expression
ovoliunlem1 (𝜑 → (𝑈𝐾) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Distinct variable groups:   𝐴,𝑘   𝑘,𝑛,𝐵   𝑘,𝐹,𝑛   𝑤,𝑘,𝐽,𝑛   𝑛,𝐾,𝑤   𝑘,𝐿,𝑛,𝑤   𝑛,𝐻   𝜑,𝑘,𝑛   𝑆,𝑘   𝑘,𝐺   𝑇,𝑘   𝑛,𝐺   𝑇,𝑛
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤,𝑛)   𝐵(𝑤)   𝑆(𝑤,𝑛)   𝑇(𝑤)   𝑈(𝑤,𝑘,𝑛)   𝐹(𝑤)   𝐺(𝑤)   𝐻(𝑤,𝑘)   𝐾(𝑘)

Proof of Theorem ovoliunlem1
Dummy variables 𝑗 𝑚 𝑥 𝑦 𝑧 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . . . . . . 9 (𝑗 = (𝐽𝑚) → (1st𝑗) = (1st ‘(𝐽𝑚)))
21fveq2d 6233 . . . . . . . 8 (𝑗 = (𝐽𝑚) → (𝐹‘(1st𝑗)) = (𝐹‘(1st ‘(𝐽𝑚))))
3 fveq2 6229 . . . . . . . 8 (𝑗 = (𝐽𝑚) → (2nd𝑗) = (2nd ‘(𝐽𝑚)))
42, 3fveq12d 6235 . . . . . . 7 (𝑗 = (𝐽𝑚) → ((𝐹‘(1st𝑗))‘(2nd𝑗)) = ((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))
54fveq2d 6233 . . . . . 6 (𝑗 = (𝐽𝑚) → (2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) = (2nd ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))))
64fveq2d 6233 . . . . . 6 (𝑗 = (𝐽𝑚) → (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) = (1st ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))))
75, 6oveq12d 6708 . . . . 5 (𝑗 = (𝐽𝑚) → ((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) = ((2nd ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))) − (1st ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))))
8 fzfid 12812 . . . . 5 (𝜑 → (1...𝐾) ∈ Fin)
9 ovoliun.j . . . . . . 7 (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))
10 f1of1 6174 . . . . . . 7 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ–1-1→(ℕ × ℕ))
119, 10syl 17 . . . . . 6 (𝜑𝐽:ℕ–1-1→(ℕ × ℕ))
12 elfznn 12408 . . . . . . 7 (𝑚 ∈ (1...𝐾) → 𝑚 ∈ ℕ)
1312ssriv 3640 . . . . . 6 (1...𝐾) ⊆ ℕ
14 f1ores 6189 . . . . . 6 ((𝐽:ℕ–1-1→(ℕ × ℕ) ∧ (1...𝐾) ⊆ ℕ) → (𝐽 ↾ (1...𝐾)):(1...𝐾)–1-1-onto→(𝐽 “ (1...𝐾)))
1511, 13, 14sylancl 695 . . . . 5 (𝜑 → (𝐽 ↾ (1...𝐾)):(1...𝐾)–1-1-onto→(𝐽 “ (1...𝐾)))
16 fvres 6245 . . . . . 6 (𝑚 ∈ (1...𝐾) → ((𝐽 ↾ (1...𝐾))‘𝑚) = (𝐽𝑚))
1716adantl 481 . . . . 5 ((𝜑𝑚 ∈ (1...𝐾)) → ((𝐽 ↾ (1...𝐾))‘𝑚) = (𝐽𝑚))
18 inss2 3867 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
19 ovoliun.f . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
2019adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
21 imassrn 5512 . . . . . . . . . . . . . . 15 (𝐽 “ (1...𝐾)) ⊆ ran 𝐽
22 f1of 6175 . . . . . . . . . . . . . . . . 17 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ⟶(ℕ × ℕ))
239, 22syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐽:ℕ⟶(ℕ × ℕ))
24 frn 6091 . . . . . . . . . . . . . . . 16 (𝐽:ℕ⟶(ℕ × ℕ) → ran 𝐽 ⊆ (ℕ × ℕ))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐽 ⊆ (ℕ × ℕ))
2621, 25syl5ss 3647 . . . . . . . . . . . . . 14 (𝜑 → (𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ))
2726sselda 3636 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝑗 ∈ (ℕ × ℕ))
28 xp1st 7242 . . . . . . . . . . . . 13 (𝑗 ∈ (ℕ × ℕ) → (1st𝑗) ∈ ℕ)
2927, 28syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (1st𝑗) ∈ ℕ)
3020, 29ffvelrnd 6400 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (𝐹‘(1st𝑗)) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
31 reex 10065 . . . . . . . . . . . . . 14 ℝ ∈ V
3231, 31xpex 7004 . . . . . . . . . . . . 13 (ℝ × ℝ) ∈ V
3332inex2 4833 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ∈ V
34 nnex 11064 . . . . . . . . . . . 12 ℕ ∈ V
3533, 34elmap 7928 . . . . . . . . . . 11 ((𝐹‘(1st𝑗)) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ (𝐹‘(1st𝑗)):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3630, 35sylib 208 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (𝐹‘(1st𝑗)):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
37 xp2nd 7243 . . . . . . . . . . 11 (𝑗 ∈ (ℕ × ℕ) → (2nd𝑗) ∈ ℕ)
3827, 37syl 17 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (2nd𝑗) ∈ ℕ)
3936, 38ffvelrnd 6400 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → ((𝐹‘(1st𝑗))‘(2nd𝑗)) ∈ ( ≤ ∩ (ℝ × ℝ)))
4018, 39sseldi 3634 . . . . . . . 8 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → ((𝐹‘(1st𝑗))‘(2nd𝑗)) ∈ (ℝ × ℝ))
41 xp2nd 7243 . . . . . . . 8 (((𝐹‘(1st𝑗))‘(2nd𝑗)) ∈ (ℝ × ℝ) → (2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) ∈ ℝ)
4240, 41syl 17 . . . . . . 7 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) ∈ ℝ)
43 xp1st 7242 . . . . . . . 8 (((𝐹‘(1st𝑗))‘(2nd𝑗)) ∈ (ℝ × ℝ) → (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) ∈ ℝ)
4440, 43syl 17 . . . . . . 7 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) ∈ ℝ)
4542, 44resubcld 10496 . . . . . 6 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → ((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) ∈ ℝ)
4645recnd 10106 . . . . 5 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → ((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) ∈ ℂ)
477, 8, 15, 17, 46fsumf1o 14498 . . . 4 (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) = Σ𝑚 ∈ (1...𝐾)((2nd ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))) − (1st ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))))
4819adantr 480 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
4923ffvelrnda 6399 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐽𝑘) ∈ (ℕ × ℕ))
50 xp1st 7242 . . . . . . . . . . . 12 ((𝐽𝑘) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
5149, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
5248, 51ffvelrnd 6400 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
5333, 34elmap 7928 . . . . . . . . . 10 ((𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ (𝐹‘(1st ‘(𝐽𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
5452, 53sylib 208 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
55 xp2nd 7243 . . . . . . . . . 10 ((𝐽𝑘) ∈ (ℕ × ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
5649, 55syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
5754, 56ffvelrnd 6400 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) ∈ ( ≤ ∩ (ℝ × ℝ)))
58 ovoliun.h . . . . . . . 8 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))
5957, 58fmptd 6425 . . . . . . 7 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
60 eqid 2651 . . . . . . . 8 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
6160ovolfsval 23285 . . . . . . 7 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑚) = ((2nd ‘(𝐻𝑚)) − (1st ‘(𝐻𝑚))))
6259, 12, 61syl2an 493 . . . . . 6 ((𝜑𝑚 ∈ (1...𝐾)) → (((abs ∘ − ) ∘ 𝐻)‘𝑚) = ((2nd ‘(𝐻𝑚)) − (1st ‘(𝐻𝑚))))
6312adantl 481 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...𝐾)) → 𝑚 ∈ ℕ)
64 fveq2 6229 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (𝐽𝑘) = (𝐽𝑚))
6564fveq2d 6233 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (1st ‘(𝐽𝑘)) = (1st ‘(𝐽𝑚)))
6665fveq2d 6233 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝐹‘(1st ‘(𝐽𝑘))) = (𝐹‘(1st ‘(𝐽𝑚))))
6764fveq2d 6233 . . . . . . . . . . 11 (𝑘 = 𝑚 → (2nd ‘(𝐽𝑘)) = (2nd ‘(𝐽𝑚)))
6866, 67fveq12d 6235 . . . . . . . . . 10 (𝑘 = 𝑚 → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) = ((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))
69 fvex 6239 . . . . . . . . . 10 ((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))) ∈ V
7068, 58, 69fvmpt 6321 . . . . . . . . 9 (𝑚 ∈ ℕ → (𝐻𝑚) = ((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))
7163, 70syl 17 . . . . . . . 8 ((𝜑𝑚 ∈ (1...𝐾)) → (𝐻𝑚) = ((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))
7271fveq2d 6233 . . . . . . 7 ((𝜑𝑚 ∈ (1...𝐾)) → (2nd ‘(𝐻𝑚)) = (2nd ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))))
7371fveq2d 6233 . . . . . . 7 ((𝜑𝑚 ∈ (1...𝐾)) → (1st ‘(𝐻𝑚)) = (1st ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))))
7472, 73oveq12d 6708 . . . . . 6 ((𝜑𝑚 ∈ (1...𝐾)) → ((2nd ‘(𝐻𝑚)) − (1st ‘(𝐻𝑚))) = ((2nd ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))) − (1st ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))))
7562, 74eqtrd 2685 . . . . 5 ((𝜑𝑚 ∈ (1...𝐾)) → (((abs ∘ − ) ∘ 𝐻)‘𝑚) = ((2nd ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))) − (1st ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))))
76 ovoliun.k . . . . . 6 (𝜑𝐾 ∈ ℕ)
77 nnuz 11761 . . . . . 6 ℕ = (ℤ‘1)
7876, 77syl6eleq 2740 . . . . 5 (𝜑𝐾 ∈ (ℤ‘1))
79 ffvelrn 6397 . . . . . . . . . . 11 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (𝐻𝑚) ∈ ( ≤ ∩ (ℝ × ℝ)))
8059, 12, 79syl2an 493 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝐾)) → (𝐻𝑚) ∈ ( ≤ ∩ (ℝ × ℝ)))
8118, 80sseldi 3634 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...𝐾)) → (𝐻𝑚) ∈ (ℝ × ℝ))
82 xp2nd 7243 . . . . . . . . 9 ((𝐻𝑚) ∈ (ℝ × ℝ) → (2nd ‘(𝐻𝑚)) ∈ ℝ)
8381, 82syl 17 . . . . . . . 8 ((𝜑𝑚 ∈ (1...𝐾)) → (2nd ‘(𝐻𝑚)) ∈ ℝ)
84 xp1st 7242 . . . . . . . . 9 ((𝐻𝑚) ∈ (ℝ × ℝ) → (1st ‘(𝐻𝑚)) ∈ ℝ)
8581, 84syl 17 . . . . . . . 8 ((𝜑𝑚 ∈ (1...𝐾)) → (1st ‘(𝐻𝑚)) ∈ ℝ)
8683, 85resubcld 10496 . . . . . . 7 ((𝜑𝑚 ∈ (1...𝐾)) → ((2nd ‘(𝐻𝑚)) − (1st ‘(𝐻𝑚))) ∈ ℝ)
8786recnd 10106 . . . . . 6 ((𝜑𝑚 ∈ (1...𝐾)) → ((2nd ‘(𝐻𝑚)) − (1st ‘(𝐻𝑚))) ∈ ℂ)
8874, 87eqeltrrd 2731 . . . . 5 ((𝜑𝑚 ∈ (1...𝐾)) → ((2nd ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))) − (1st ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))) ∈ ℂ)
8975, 78, 88fsumser 14505 . . . 4 (𝜑 → Σ𝑚 ∈ (1...𝐾)((2nd ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚)))) − (1st ‘((𝐹‘(1st ‘(𝐽𝑚)))‘(2nd ‘(𝐽𝑚))))) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝐾))
9047, 89eqtrd 2685 . . 3 (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝐾))
91 ovoliun.u . . . 4 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
9291fveq1i 6230 . . 3 (𝑈𝐾) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝐾)
9390, 92syl6eqr 2703 . 2 (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) = (𝑈𝐾))
94 f1oeng 8016 . . . . . . 7 (((1...𝐾) ∈ Fin ∧ (𝐽 ↾ (1...𝐾)):(1...𝐾)–1-1-onto→(𝐽 “ (1...𝐾))) → (1...𝐾) ≈ (𝐽 “ (1...𝐾)))
958, 15, 94syl2anc 694 . . . . . 6 (𝜑 → (1...𝐾) ≈ (𝐽 “ (1...𝐾)))
9695ensymd 8048 . . . . 5 (𝜑 → (𝐽 “ (1...𝐾)) ≈ (1...𝐾))
97 enfii 8218 . . . . 5 (((1...𝐾) ∈ Fin ∧ (𝐽 “ (1...𝐾)) ≈ (1...𝐾)) → (𝐽 “ (1...𝐾)) ∈ Fin)
988, 96, 97syl2anc 694 . . . 4 (𝜑 → (𝐽 “ (1...𝐾)) ∈ Fin)
9998, 45fsumrecl 14509 . . 3 (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) ∈ ℝ)
100 fzfid 12812 . . . . 5 (𝜑 → (1...𝐿) ∈ Fin)
101 elfznn 12408 . . . . . 6 (𝑛 ∈ (1...𝐿) → 𝑛 ∈ ℕ)
102 ovoliun.v . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
103101, 102sylan2 490 . . . . 5 ((𝜑𝑛 ∈ (1...𝐿)) → (vol*‘𝐴) ∈ ℝ)
104100, 103fsumrecl 14509 . . . 4 (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ∈ ℝ)
105 ovoliun.b . . . . . . 7 (𝜑𝐵 ∈ ℝ+)
106105rpred 11910 . . . . . 6 (𝜑𝐵 ∈ ℝ)
107 2nn 11223 . . . . . . . 8 2 ∈ ℕ
108 nnnn0 11337 . . . . . . . 8 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
109 nnexpcl 12913 . . . . . . . 8 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
110107, 108, 109sylancr 696 . . . . . . 7 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
111101, 110syl 17 . . . . . 6 (𝑛 ∈ (1...𝐿) → (2↑𝑛) ∈ ℕ)
112 nndivre 11094 . . . . . 6 ((𝐵 ∈ ℝ ∧ (2↑𝑛) ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ)
113106, 111, 112syl2an 493 . . . . 5 ((𝜑𝑛 ∈ (1...𝐿)) → (𝐵 / (2↑𝑛)) ∈ ℝ)
114100, 113fsumrecl 14509 . . . 4 (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) ∈ ℝ)
115104, 114readdcld 10107 . . 3 (𝜑 → (Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) + Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛))) ∈ ℝ)
116 ovoliun.r . . . 4 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
117116, 106readdcld 10107 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ)
118 relxp 5160 . . . . . . . . . . . . . . 15 Rel ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))
119 relres 5461 . . . . . . . . . . . . . . 15 Rel ((𝐽 “ (1...𝐾)) ↾ {𝑛})
120 opelxp 5180 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝑦⟩ ∈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ↔ (𝑥 ∈ {𝑛} ∧ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
121 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
122121opelres 5436 . . . . . . . . . . . . . . . . 17 (⟨𝑥, 𝑦⟩ ∈ ((𝐽 “ (1...𝐾)) ↾ {𝑛}) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) ∧ 𝑥 ∈ {𝑛}))
123 ancom 465 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ {𝑛} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) ∧ 𝑥 ∈ {𝑛}))
124 elsni 4227 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝑛} → 𝑥 = 𝑛)
125124opeq1d 4439 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑛} → ⟨𝑥, 𝑦⟩ = ⟨𝑛, 𝑦⟩)
126125eleq1d 2715 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ {𝑛} → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) ↔ ⟨𝑛, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))))
127 vex 3234 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ V
128127, 121elimasn 5525 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}) ↔ ⟨𝑛, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)))
129126, 128syl6bbr 278 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ {𝑛} → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) ↔ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
130129pm5.32i 670 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ {𝑛} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))) ↔ (𝑥 ∈ {𝑛} ∧ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
131122, 123, 1303bitr2ri 289 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ {𝑛} ∧ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐽 “ (1...𝐾)) ↾ {𝑛}))
132120, 131bitri 264 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐽 “ (1...𝐾)) ↾ {𝑛}))
133118, 119, 132eqrelriiv 5248 . . . . . . . . . . . . . 14 ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ↾ {𝑛})
134 df-res 5155 . . . . . . . . . . . . . 14 ((𝐽 “ (1...𝐾)) ↾ {𝑛}) = ((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V))
135133, 134eqtri 2673 . . . . . . . . . . . . 13 ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V))
136135a1i 11 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝐿)) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V)))
137136iuneq2dv 4574 . . . . . . . . . . 11 (𝜑 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = 𝑛 ∈ (1...𝐿)((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V)))
138 iunin2 4616 . . . . . . . . . . 11 𝑛 ∈ (1...𝐿)((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V)) = ((𝐽 “ (1...𝐾)) ∩ 𝑛 ∈ (1...𝐿)({𝑛} × V))
139137, 138syl6eq 2701 . . . . . . . . . 10 (𝜑 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ∩ 𝑛 ∈ (1...𝐿)({𝑛} × V)))
140 relxp 5160 . . . . . . . . . . . . . 14 Rel (ℕ × ℕ)
141 relss 5240 . . . . . . . . . . . . . 14 ((𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ) → (Rel (ℕ × ℕ) → Rel (𝐽 “ (1...𝐾))))
14226, 140, 141mpisyl 21 . . . . . . . . . . . . 13 (𝜑 → Rel (𝐽 “ (1...𝐾)))
143 ovoliun.l2 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽𝑤)) ≤ 𝐿)
144 ffn 6083 . . . . . . . . . . . . . . . . . . . . 21 (𝐽:ℕ⟶(ℕ × ℕ) → 𝐽 Fn ℕ)
14523, 144syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 Fn ℕ)
146 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = (𝐽𝑤) → (1st𝑗) = (1st ‘(𝐽𝑤)))
147146breq1d 4695 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝐽𝑤) → ((1st𝑗) ≤ 𝐿 ↔ (1st ‘(𝐽𝑤)) ≤ 𝐿))
148147ralima 6538 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 Fn ℕ ∧ (1...𝐾) ⊆ ℕ) → (∀𝑗 ∈ (𝐽 “ (1...𝐾))(1st𝑗) ≤ 𝐿 ↔ ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽𝑤)) ≤ 𝐿))
149145, 13, 148sylancl 695 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∀𝑗 ∈ (𝐽 “ (1...𝐾))(1st𝑗) ≤ 𝐿 ↔ ∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽𝑤)) ≤ 𝐿))
150143, 149mpbird 247 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑗 ∈ (𝐽 “ (1...𝐾))(1st𝑗) ≤ 𝐿)
151150r19.21bi 2961 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (1st𝑗) ≤ 𝐿)
15229, 77syl6eleq 2740 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (1st𝑗) ∈ (ℤ‘1))
153 ovoliun.l1 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐿 ∈ ℤ)
154153adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝐿 ∈ ℤ)
155 elfz5 12372 . . . . . . . . . . . . . . . . . 18 (((1st𝑗) ∈ (ℤ‘1) ∧ 𝐿 ∈ ℤ) → ((1st𝑗) ∈ (1...𝐿) ↔ (1st𝑗) ≤ 𝐿))
156152, 154, 155syl2anc 694 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → ((1st𝑗) ∈ (1...𝐿) ↔ (1st𝑗) ≤ 𝐿))
157151, 156mpbird 247 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝐽 “ (1...𝐾))) → (1st𝑗) ∈ (1...𝐿))
158157ralrimiva 2995 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑗 ∈ (𝐽 “ (1...𝐾))(1st𝑗) ∈ (1...𝐿))
159 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
160159, 121op1std 7220 . . . . . . . . . . . . . . . . 17 (𝑗 = ⟨𝑥, 𝑦⟩ → (1st𝑗) = 𝑥)
161160eleq1d 2715 . . . . . . . . . . . . . . . 16 (𝑗 = ⟨𝑥, 𝑦⟩ → ((1st𝑗) ∈ (1...𝐿) ↔ 𝑥 ∈ (1...𝐿)))
162161rspccv 3337 . . . . . . . . . . . . . . 15 (∀𝑗 ∈ (𝐽 “ (1...𝐾))(1st𝑗) ∈ (1...𝐿) → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) → 𝑥 ∈ (1...𝐿)))
163158, 162syl 17 . . . . . . . . . . . . . 14 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) → 𝑥 ∈ (1...𝐿)))
164 opelxp 5180 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ ( 𝑛 ∈ (1...𝐿){𝑛} × V) ↔ (𝑥 𝑛 ∈ (1...𝐿){𝑛} ∧ 𝑦 ∈ V))
165121biantru 525 . . . . . . . . . . . . . . 15 (𝑥 𝑛 ∈ (1...𝐿){𝑛} ↔ (𝑥 𝑛 ∈ (1...𝐿){𝑛} ∧ 𝑦 ∈ V))
166 iunid 4607 . . . . . . . . . . . . . . . 16 𝑛 ∈ (1...𝐿){𝑛} = (1...𝐿)
167166eleq2i 2722 . . . . . . . . . . . . . . 15 (𝑥 𝑛 ∈ (1...𝐿){𝑛} ↔ 𝑥 ∈ (1...𝐿))
168164, 165, 1673bitr2i 288 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑦⟩ ∈ ( 𝑛 ∈ (1...𝐿){𝑛} × V) ↔ 𝑥 ∈ (1...𝐿))
169163, 168syl6ibr 242 . . . . . . . . . . . . 13 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) → ⟨𝑥, 𝑦⟩ ∈ ( 𝑛 ∈ (1...𝐿){𝑛} × V)))
170142, 169relssdv 5246 . . . . . . . . . . . 12 (𝜑 → (𝐽 “ (1...𝐾)) ⊆ ( 𝑛 ∈ (1...𝐿){𝑛} × V))
171 xpiundir 5208 . . . . . . . . . . . 12 ( 𝑛 ∈ (1...𝐿){𝑛} × V) = 𝑛 ∈ (1...𝐿)({𝑛} × V)
172170, 171syl6sseq 3684 . . . . . . . . . . 11 (𝜑 → (𝐽 “ (1...𝐾)) ⊆ 𝑛 ∈ (1...𝐿)({𝑛} × V))
173 df-ss 3621 . . . . . . . . . . 11 ((𝐽 “ (1...𝐾)) ⊆ 𝑛 ∈ (1...𝐿)({𝑛} × V) ↔ ((𝐽 “ (1...𝐾)) ∩ 𝑛 ∈ (1...𝐿)({𝑛} × V)) = (𝐽 “ (1...𝐾)))
174172, 173sylib 208 . . . . . . . . . 10 (𝜑 → ((𝐽 “ (1...𝐾)) ∩ 𝑛 ∈ (1...𝐿)({𝑛} × V)) = (𝐽 “ (1...𝐾)))
175139, 174eqtrd 2685 . . . . . . . . 9 (𝜑 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = (𝐽 “ (1...𝐾)))
176175, 98eqeltrd 2730 . . . . . . . 8 (𝜑 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin)
177 ssiun2 4595 . . . . . . . 8 (𝑛 ∈ (1...𝐿) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ⊆ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})))
178 ssfi 8221 . . . . . . . 8 (( 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin ∧ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ⊆ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin)
179176, 177, 178syl2an 493 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝐿)) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin)
180 2ndconst 7311 . . . . . . . . . 10 (𝑛 ∈ V → (2nd ↾ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))):({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))–1-1-onto→((𝐽 “ (1...𝐾)) “ {𝑛}))
181127, 180ax-mp 5 . . . . . . . . 9 (2nd ↾ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))):({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))–1-1-onto→((𝐽 “ (1...𝐾)) “ {𝑛})
182 f1oeng 8016 . . . . . . . . 9 ((({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin ∧ (2nd ↾ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))):({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))–1-1-onto→((𝐽 “ (1...𝐾)) “ {𝑛})) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ≈ ((𝐽 “ (1...𝐾)) “ {𝑛}))
183179, 181, 182sylancl 695 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ≈ ((𝐽 “ (1...𝐾)) “ {𝑛}))
184183ensymd 8048 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ≈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})))
185 enfii 8218 . . . . . . 7 ((({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin ∧ ((𝐽 “ (1...𝐾)) “ {𝑛}) ≈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ∈ Fin)
186179, 184, 185syl2anc 694 . . . . . 6 ((𝜑𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ∈ Fin)
187 ffvelrn 6397 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
18819, 101, 187syl2an 493 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝐿)) → (𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
18933, 34elmap 7928 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
190188, 189sylib 208 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝐿)) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
191190adantrr 753 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
192 imassrn 5512 . . . . . . . . . . . . . 14 ((𝐽 “ (1...𝐾)) “ {𝑛}) ⊆ ran (𝐽 “ (1...𝐾))
19326adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝐿)) → (𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ))
194 rnss 5386 . . . . . . . . . . . . . . . 16 ((𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ) → ran (𝐽 “ (1...𝐾)) ⊆ ran (ℕ × ℕ))
195193, 194syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝐿)) → ran (𝐽 “ (1...𝐾)) ⊆ ran (ℕ × ℕ))
196 rnxpid 5602 . . . . . . . . . . . . . . 15 ran (ℕ × ℕ) = ℕ
197195, 196syl6sseq 3684 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝐿)) → ran (𝐽 “ (1...𝐾)) ⊆ ℕ)
198192, 197syl5ss 3647 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ⊆ ℕ)
199198sseld 3635 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝐿)) → (𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}) → 𝑖 ∈ ℕ))
200199impr 648 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → 𝑖 ∈ ℕ)
201191, 200ffvelrnd 6400 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((𝐹𝑛)‘𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
20218, 201sseldi 3634 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((𝐹𝑛)‘𝑖) ∈ (ℝ × ℝ))
203 xp2nd 7243 . . . . . . . . 9 (((𝐹𝑛)‘𝑖) ∈ (ℝ × ℝ) → (2nd ‘((𝐹𝑛)‘𝑖)) ∈ ℝ)
204202, 203syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → (2nd ‘((𝐹𝑛)‘𝑖)) ∈ ℝ)
205 xp1st 7242 . . . . . . . . 9 (((𝐹𝑛)‘𝑖) ∈ (ℝ × ℝ) → (1st ‘((𝐹𝑛)‘𝑖)) ∈ ℝ)
206202, 205syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → (1st ‘((𝐹𝑛)‘𝑖)) ∈ ℝ)
207204, 206resubcld 10496 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ ℝ)
208207anassrs 681 . . . . . 6 (((𝜑𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})) → ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ ℝ)
209186, 208fsumrecl 14509 . . . . 5 ((𝜑𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ ℝ)
210106, 110, 112syl2an 493 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ)
211102, 210readdcld 10107 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ∈ ℝ)
212101, 211sylan2 490 . . . . 5 ((𝜑𝑛 ∈ (1...𝐿)) → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ∈ ℝ)
213 eqid 2651 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ (𝐹𝑛)) = ((abs ∘ − ) ∘ (𝐹𝑛))
214 ovoliun.s . . . . . . . . . . . 12 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))
215213, 214ovolsf 23287 . . . . . . . . . . 11 ((𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
216190, 215syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝐿)) → 𝑆:ℕ⟶(0[,)+∞))
217 frn 6091 . . . . . . . . . 10 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
218216, 217syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ (0[,)+∞))
219 icossxr 12296 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ*
220218, 219syl6ss 3648 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ ℝ*)
221 supxrcl 12183 . . . . . . . 8 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
222220, 221syl 17 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
223 mnfxr 10134 . . . . . . . . 9 -∞ ∈ ℝ*
224223a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → -∞ ∈ ℝ*)
225103rexrd 10127 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → (vol*‘𝐴) ∈ ℝ*)
226 mnflt 11995 . . . . . . . . 9 ((vol*‘𝐴) ∈ ℝ → -∞ < (vol*‘𝐴))
227103, 226syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → -∞ < (vol*‘𝐴))
228 ovoliun.x1 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
229101, 228sylan2 490 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝐿)) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
230214ovollb 23293 . . . . . . . . 9 (((𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ (𝐹𝑛))) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
231190, 229, 230syl2anc 694 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
232224, 225, 222, 227, 231xrltletrd 12030 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝐿)) → -∞ < sup(ran 𝑆, ℝ*, < ))
233 ovoliun.x2 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
234101, 233sylan2 490 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
235 xrre 12038 . . . . . . 7 (((sup(ran 𝑆, ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ∈ ℝ) ∧ (-∞ < sup(ran 𝑆, ℝ*, < ) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
236222, 212, 232, 234, 235syl22anc 1367 . . . . . 6 ((𝜑𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
237 1zzd 11446 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → 1 ∈ ℤ)
238213ovolfsval 23285 . . . . . . . . 9 (((𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑖 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐹𝑛))‘𝑖) = ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))))
239190, 238sylan 487 . . . . . . . 8 (((𝜑𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐹𝑛))‘𝑖) = ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))))
240213ovolfsf 23286 . . . . . . . . . . . . 13 ((𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ (𝐹𝑛)):ℕ⟶(0[,)+∞))
241190, 240syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝐿)) → ((abs ∘ − ) ∘ (𝐹𝑛)):ℕ⟶(0[,)+∞))
242241ffvelrnda 6399 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐹𝑛))‘𝑖) ∈ (0[,)+∞))
243239, 242eqeltrrd 2731 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ (0[,)+∞))
244 elrege0 12316 . . . . . . . . . 10 (((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ (0[,)+∞) ↔ (((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ ℝ ∧ 0 ≤ ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖)))))
245243, 244sylib 208 . . . . . . . . 9 (((𝜑𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → (((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ ℝ ∧ 0 ≤ ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖)))))
246245simpld 474 . . . . . . . 8 (((𝜑𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ ℝ)
247245simprd 478 . . . . . . . 8 (((𝜑𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → 0 ≤ ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))))
248 supxrub 12192 . . . . . . . . . . . . . . 15 ((ran 𝑆 ⊆ ℝ*𝑧 ∈ ran 𝑆) → 𝑧 ≤ sup(ran 𝑆, ℝ*, < ))
249220, 248sylan 487 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (1...𝐿)) ∧ 𝑧 ∈ ran 𝑆) → 𝑧 ≤ sup(ran 𝑆, ℝ*, < ))
250249ralrimiva 2995 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝐿)) → ∀𝑧 ∈ ran 𝑆 𝑧 ≤ sup(ran 𝑆, ℝ*, < ))
251 breq2 4689 . . . . . . . . . . . . . . 15 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (𝑧𝑥𝑧 ≤ sup(ran 𝑆, ℝ*, < )))
252251ralbidv 3015 . . . . . . . . . . . . . 14 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ sup(ran 𝑆, ℝ*, < )))
253252rspcev 3340 . . . . . . . . . . . . 13 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥)
254236, 250, 253syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝐿)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥)
255 ffn 6083 . . . . . . . . . . . . . . 15 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
256216, 255syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝐿)) → 𝑆 Fn ℕ)
257 breq1 4688 . . . . . . . . . . . . . . 15 (𝑧 = (𝑆𝑘) → (𝑧𝑥 ↔ (𝑆𝑘) ≤ 𝑥))
258257ralrn 6402 . . . . . . . . . . . . . 14 (𝑆 Fn ℕ → (∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
259256, 258syl 17 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝐿)) → (∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
260259rexbidv 3081 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝐿)) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥))
261254, 260mpbid 222 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝐿)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ 𝑥)
26277, 214, 237, 239, 246, 247, 261isumsup2 14622 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝐿)) → 𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
263214, 262syl5eqbrr 4721 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝐿)) → seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛))) ⇝ sup(ran 𝑆, ℝ, < ))
264 climrel 14267 . . . . . . . . . 10 Rel ⇝
265264releldmi 5394 . . . . . . . . 9 (seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛))) ⇝ sup(ran 𝑆, ℝ, < ) → seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛))) ∈ dom ⇝ )
266263, 265syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛))) ∈ dom ⇝ )
26777, 237, 186, 198, 239, 246, 247, 266isumless 14621 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ≤ Σ𝑖 ∈ ℕ ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))))
26877, 214, 237, 239, 246, 247, 261isumsup 14623 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ℕ ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) = sup(ran 𝑆, ℝ, < ))
269 rge0ssre 12318 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ
270218, 269syl6ss 3648 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ ℝ)
271 1nn 11069 . . . . . . . . . . . 12 1 ∈ ℕ
272 fdm 6089 . . . . . . . . . . . . 13 (𝑆:ℕ⟶(0[,)+∞) → dom 𝑆 = ℕ)
273216, 272syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝐿)) → dom 𝑆 = ℕ)
274271, 273syl5eleqr 2737 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝐿)) → 1 ∈ dom 𝑆)
275 ne0i 3954 . . . . . . . . . . 11 (1 ∈ dom 𝑆 → dom 𝑆 ≠ ∅)
276274, 275syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝐿)) → dom 𝑆 ≠ ∅)
277 dm0rn0 5374 . . . . . . . . . . 11 (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
278277necon3bii 2875 . . . . . . . . . 10 (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
279276, 278sylib 208 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝐿)) → ran 𝑆 ≠ ∅)
280 supxrre 12195 . . . . . . . . 9 ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧𝑥) → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < ))
281270, 279, 254, 280syl3anc 1366 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ*, < ) = sup(ran 𝑆, ℝ, < ))
282268, 281eqtr4d 2688 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ℕ ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) = sup(ran 𝑆, ℝ*, < ))
283267, 282breqtrd 4711 . . . . . 6 ((𝜑𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ≤ sup(ran 𝑆, ℝ*, < ))
284209, 236, 212, 283, 234letrd 10232 . . . . 5 ((𝜑𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
285100, 209, 212, 284fsumle 14575 . . . 4 (𝜑 → Σ𝑛 ∈ (1...𝐿𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ≤ Σ𝑛 ∈ (1...𝐿)((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
286 vex 3234 . . . . . . . . . . 11 𝑖 ∈ V
287127, 286op1std 7220 . . . . . . . . . 10 (𝑗 = ⟨𝑛, 𝑖⟩ → (1st𝑗) = 𝑛)
288287fveq2d 6233 . . . . . . . . 9 (𝑗 = ⟨𝑛, 𝑖⟩ → (𝐹‘(1st𝑗)) = (𝐹𝑛))
289127, 286op2ndd 7221 . . . . . . . . 9 (𝑗 = ⟨𝑛, 𝑖⟩ → (2nd𝑗) = 𝑖)
290288, 289fveq12d 6235 . . . . . . . 8 (𝑗 = ⟨𝑛, 𝑖⟩ → ((𝐹‘(1st𝑗))‘(2nd𝑗)) = ((𝐹𝑛)‘𝑖))
291290fveq2d 6233 . . . . . . 7 (𝑗 = ⟨𝑛, 𝑖⟩ → (2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) = (2nd ‘((𝐹𝑛)‘𝑖)))
292290fveq2d 6233 . . . . . . 7 (𝑗 = ⟨𝑛, 𝑖⟩ → (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) = (1st ‘((𝐹𝑛)‘𝑖)))
293291, 292oveq12d 6708 . . . . . 6 (𝑗 = ⟨𝑛, 𝑖⟩ → ((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) = ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))))
294207recnd 10106 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) ∈ ℂ)
295293, 100, 186, 294fsum2d 14546 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...𝐿𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) = Σ𝑗 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))))
296175sumeq1d 14475 . . . . 5 (𝜑 → Σ𝑗 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) = Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))))
297295, 296eqtrd 2685 . . . 4 (𝜑 → Σ𝑛 ∈ (1...𝐿𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2nd ‘((𝐹𝑛)‘𝑖)) − (1st ‘((𝐹𝑛)‘𝑖))) = Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))))
298103recnd 10106 . . . . 5 ((𝜑𝑛 ∈ (1...𝐿)) → (vol*‘𝐴) ∈ ℂ)
299113recnd 10106 . . . . 5 ((𝜑𝑛 ∈ (1...𝐿)) → (𝐵 / (2↑𝑛)) ∈ ℂ)
300100, 298, 299fsumadd 14514 . . . 4 (𝜑 → Σ𝑛 ∈ (1...𝐿)((vol*‘𝐴) + (𝐵 / (2↑𝑛))) = (Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) + Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛))))
301285, 297, 3003brtr3d 4716 . . 3 (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) ≤ (Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) + Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛))))
302 1zzd 11446 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
303 simpr 476 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
304 ovoliun.g . . . . . . . . . . . 12 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
305304fvmpt2 6330 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐺𝑛) = (vol*‘𝐴))
306303, 102, 305syl2anc 694 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = (vol*‘𝐴))
307306, 102eqeltrd 2730 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
30877, 302, 307serfre 12870 . . . . . . . 8 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
309 ovoliun.t . . . . . . . . 9 𝑇 = seq1( + , 𝐺)
310309feq1i 6074 . . . . . . . 8 (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
311308, 310sylibr 224 . . . . . . 7 (𝜑𝑇:ℕ⟶ℝ)
312 frn 6091 . . . . . . 7 (𝑇:ℕ⟶ℝ → ran 𝑇 ⊆ ℝ)
313311, 312syl 17 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ)
314 ressxr 10121 . . . . . 6 ℝ ⊆ ℝ*
315313, 314syl6ss 3648 . . . . 5 (𝜑 → ran 𝑇 ⊆ ℝ*)
316101, 306sylan2 490 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝐿)) → (𝐺𝑛) = (vol*‘𝐴))
317 1red 10093 . . . . . . . . . 10 (𝜑 → 1 ∈ ℝ)
318 ffvelrn 6397 . . . . . . . . . . . . 13 ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 1 ∈ ℕ) → (𝐽‘1) ∈ (ℕ × ℕ))
31923, 271, 318sylancl 695 . . . . . . . . . . . 12 (𝜑 → (𝐽‘1) ∈ (ℕ × ℕ))
320 xp1st 7242 . . . . . . . . . . . 12 ((𝐽‘1) ∈ (ℕ × ℕ) → (1st ‘(𝐽‘1)) ∈ ℕ)
321319, 320syl 17 . . . . . . . . . . 11 (𝜑 → (1st ‘(𝐽‘1)) ∈ ℕ)
322321nnred 11073 . . . . . . . . . 10 (𝜑 → (1st ‘(𝐽‘1)) ∈ ℝ)
323153zred 11520 . . . . . . . . . 10 (𝜑𝐿 ∈ ℝ)
324321nnge1d 11101 . . . . . . . . . 10 (𝜑 → 1 ≤ (1st ‘(𝐽‘1)))
325 eluzfz1 12386 . . . . . . . . . . . 12 (𝐾 ∈ (ℤ‘1) → 1 ∈ (1...𝐾))
32678, 325syl 17 . . . . . . . . . . 11 (𝜑 → 1 ∈ (1...𝐾))
327 fveq2 6229 . . . . . . . . . . . . . 14 (𝑤 = 1 → (𝐽𝑤) = (𝐽‘1))
328327fveq2d 6233 . . . . . . . . . . . . 13 (𝑤 = 1 → (1st ‘(𝐽𝑤)) = (1st ‘(𝐽‘1)))
329328breq1d 4695 . . . . . . . . . . . 12 (𝑤 = 1 → ((1st ‘(𝐽𝑤)) ≤ 𝐿 ↔ (1st ‘(𝐽‘1)) ≤ 𝐿))
330329rspcv 3336 . . . . . . . . . . 11 (1 ∈ (1...𝐾) → (∀𝑤 ∈ (1...𝐾)(1st ‘(𝐽𝑤)) ≤ 𝐿 → (1st ‘(𝐽‘1)) ≤ 𝐿))
331326, 143, 330sylc 65 . . . . . . . . . 10 (𝜑 → (1st ‘(𝐽‘1)) ≤ 𝐿)
332317, 322, 323, 324, 331letrd 10232 . . . . . . . . 9 (𝜑 → 1 ≤ 𝐿)
333 elnnz1 11441 . . . . . . . . 9 (𝐿 ∈ ℕ ↔ (𝐿 ∈ ℤ ∧ 1 ≤ 𝐿))
334153, 332, 333sylanbrc 699 . . . . . . . 8 (𝜑𝐿 ∈ ℕ)
335334, 77syl6eleq 2740 . . . . . . 7 (𝜑𝐿 ∈ (ℤ‘1))
336316, 335, 298fsumser 14505 . . . . . 6 (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) = (seq1( + , 𝐺)‘𝐿))
337 seqfn 12853 . . . . . . . . 9 (1 ∈ ℤ → seq1( + , 𝐺) Fn (ℤ‘1))
338302, 337syl 17 . . . . . . . 8 (𝜑 → seq1( + , 𝐺) Fn (ℤ‘1))
339 fnfvelrn 6396 . . . . . . . 8 ((seq1( + , 𝐺) Fn (ℤ‘1) ∧ 𝐿 ∈ (ℤ‘1)) → (seq1( + , 𝐺)‘𝐿) ∈ ran seq1( + , 𝐺))
340338, 335, 339syl2anc 694 . . . . . . 7 (𝜑 → (seq1( + , 𝐺)‘𝐿) ∈ ran seq1( + , 𝐺))
341309rneqi 5384 . . . . . . 7 ran 𝑇 = ran seq1( + , 𝐺)
342340, 341syl6eleqr 2741 . . . . . 6 (𝜑 → (seq1( + , 𝐺)‘𝐿) ∈ ran 𝑇)
343336, 342eqeltrd 2730 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ∈ ran 𝑇)
344 supxrub 12192 . . . . 5 ((ran 𝑇 ⊆ ℝ* ∧ Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ∈ ran 𝑇) → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
345315, 343, 344syl2anc 694 . . . 4 (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
346106recnd 10106 . . . . . 6 (𝜑𝐵 ∈ ℂ)
347 geo2sum 14648 . . . . . 6 ((𝐿 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) = (𝐵 − (𝐵 / (2↑𝐿))))
348334, 346, 347syl2anc 694 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) = (𝐵 − (𝐵 / (2↑𝐿))))
349334nnnn0d 11389 . . . . . . . . . 10 (𝜑𝐿 ∈ ℕ0)
350 nnexpcl 12913 . . . . . . . . . 10 ((2 ∈ ℕ ∧ 𝐿 ∈ ℕ0) → (2↑𝐿) ∈ ℕ)
351107, 349, 350sylancr 696 . . . . . . . . 9 (𝜑 → (2↑𝐿) ∈ ℕ)
352351nnrpd 11908 . . . . . . . 8 (𝜑 → (2↑𝐿) ∈ ℝ+)
353105, 352rpdivcld 11927 . . . . . . 7 (𝜑 → (𝐵 / (2↑𝐿)) ∈ ℝ+)
354353rpge0d 11914 . . . . . 6 (𝜑 → 0 ≤ (𝐵 / (2↑𝐿)))
355106, 351nndivred 11107 . . . . . . 7 (𝜑 → (𝐵 / (2↑𝐿)) ∈ ℝ)
356106, 355subge02d 10657 . . . . . 6 (𝜑 → (0 ≤ (𝐵 / (2↑𝐿)) ↔ (𝐵 − (𝐵 / (2↑𝐿))) ≤ 𝐵))
357354, 356mpbid 222 . . . . 5 (𝜑 → (𝐵 − (𝐵 / (2↑𝐿))) ≤ 𝐵)
358348, 357eqbrtrd 4707 . . . 4 (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) ≤ 𝐵)
359104, 114, 116, 106, 345, 358le2addd 10684 . . 3 (𝜑 → (Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) + Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛))) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
36099, 115, 117, 301, 359letrd 10232 . 2 (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2nd ‘((𝐹‘(1st𝑗))‘(2nd𝑗))) − (1st ‘((𝐹‘(1st𝑗))‘(2nd𝑗)))) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
36193, 360eqbrtrrd 4709 1 (𝜑 → (𝑈𝐾) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  c0 3948  {csn 4210  cop 4216   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  ccom 5147  Rel wrel 5148   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  cen 7994  Fincfn 7997  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  +crp 11870  (,)cioo 12213  [,)cico 12215  ...cfz 12364  seqcseq 12841  cexp 12900  abscabs 14018  cli 14259  Σcsu 14460  vol*covol 23277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ioo 12217  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-ovol 23279
This theorem is referenced by:  ovoliunlem2  23317
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