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Theorem ovolicc2lem4 24131
Description: Lemma for ovolicc2 24133. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by AV, 17-Sep-2020.)
Hypotheses
Ref Expression
ovolicc.1 (𝜑𝐴 ∈ ℝ)
ovolicc.2 (𝜑𝐵 ∈ ℝ)
ovolicc.3 (𝜑𝐴𝐵)
ovolicc2.4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolicc2.5 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolicc2.6 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
ovolicc2.7 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
ovolicc2.8 (𝜑𝐺:𝑈⟶ℕ)
ovolicc2.9 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
ovolicc2.10 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
ovolicc2.11 (𝜑𝐻:𝑇𝑇)
ovolicc2.12 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
ovolicc2.13 (𝜑𝐴𝐶)
ovolicc2.14 (𝜑𝐶𝑇)
ovolicc2.15 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
ovolicc2.16 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}
ovolicc2.17 𝑀 = inf(𝑊, ℝ, < )
Assertion
Ref Expression
ovolicc2lem4 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   𝑡,𝑛,𝑢,𝐴   𝐵,𝑛,𝑡,𝑢   𝑡,𝐻   𝐶,𝑛,𝑡   𝑛,𝐹,𝑡   𝑛,𝐾,𝑡,𝑢   𝑛,𝐺,𝑡   𝑛,𝑀,𝑡   𝑛,𝑊   𝜑,𝑛,𝑡   𝑇,𝑛,𝑡   𝑈,𝑛,𝑡,𝑢
Allowed substitution hints:   𝜑(𝑢)   𝐶(𝑢)   𝑆(𝑢,𝑡,𝑛)   𝑇(𝑢)   𝐹(𝑢)   𝐺(𝑢)   𝐻(𝑢,𝑛)   𝑀(𝑢)   𝑊(𝑢,𝑡)

Proof of Theorem ovolicc2lem4
Dummy variables 𝑚 𝑥 𝑦 𝑧 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arch 11891 . . . . 5 (𝑥 ∈ ℝ → ∃𝑧 ∈ ℕ 𝑥 < 𝑧)
21ad2antlr 726 . . . 4 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) → ∃𝑧 ∈ ℕ 𝑥 < 𝑧)
3 df-ima 5555 . . . . . . . . . . . . . . . 16 ((𝐺𝐾) “ (1...𝑀)) = ran ((𝐺𝐾) ↾ (1...𝑀))
4 ovolicc2.8 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺:𝑈⟶ℕ)
5 nnuz 12278 . . . . . . . . . . . . . . . . . . . . 21 ℕ = (ℤ‘1)
6 ovolicc2.15 . . . . . . . . . . . . . . . . . . . . 21 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
7 1zzd 12010 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ∈ ℤ)
8 ovolicc2.14 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐶𝑇)
9 ovolicc2.11 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐻:𝑇𝑇)
105, 6, 7, 8, 9algrf 15915 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐾:ℕ⟶𝑇)
11 ovolicc2.10 . . . . . . . . . . . . . . . . . . . . 21 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
1211ssrab3 4043 . . . . . . . . . . . . . . . . . . . 20 𝑇𝑈
13 fss 6517 . . . . . . . . . . . . . . . . . . . 20 ((𝐾:ℕ⟶𝑇𝑇𝑈) → 𝐾:ℕ⟶𝑈)
1410, 12, 13sylancl 589 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐾:ℕ⟶𝑈)
15 fco 6521 . . . . . . . . . . . . . . . . . . 19 ((𝐺:𝑈⟶ℕ ∧ 𝐾:ℕ⟶𝑈) → (𝐺𝐾):ℕ⟶ℕ)
164, 14, 15syl2anc 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐺𝐾):ℕ⟶ℕ)
17 fz1ssnn 12942 . . . . . . . . . . . . . . . . . 18 (1...𝑀) ⊆ ℕ
18 fssres 6534 . . . . . . . . . . . . . . . . . 18 (((𝐺𝐾):ℕ⟶ℕ ∧ (1...𝑀) ⊆ ℕ) → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ)
1916, 17, 18sylancl 589 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ)
2019frnd 6510 . . . . . . . . . . . . . . . 16 (𝜑 → ran ((𝐺𝐾) ↾ (1...𝑀)) ⊆ ℕ)
213, 20eqsstrid 4001 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐺𝐾) “ (1...𝑀)) ⊆ ℕ)
22 nnssre 11638 . . . . . . . . . . . . . . 15 ℕ ⊆ ℝ
2321, 22sstrdi 3965 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺𝐾) “ (1...𝑀)) ⊆ ℝ)
2423ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → ((𝐺𝐾) “ (1...𝑀)) ⊆ ℝ)
25 simpr 488 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀)))
2624, 25sseldd 3954 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ ℝ)
27 simpllr 775 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑥 ∈ ℝ)
28 nnre 11641 . . . . . . . . . . . . 13 (𝑧 ∈ ℕ → 𝑧 ∈ ℝ)
2928ad2antlr 726 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑧 ∈ ℝ)
30 lelttr 10729 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦𝑥𝑥 < 𝑧) → 𝑦 < 𝑧))
3126, 27, 29, 30syl3anc 1368 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → ((𝑦𝑥𝑥 < 𝑧) → 𝑦 < 𝑧))
3231ancomsd 469 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → ((𝑥 < 𝑧𝑦𝑥) → 𝑦 < 𝑧))
3332expdimp 456 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) ∧ 𝑥 < 𝑧) → (𝑦𝑥𝑦 < 𝑧))
3433an32s 651 . . . . . . . 8 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑥 < 𝑧) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → (𝑦𝑥𝑦 < 𝑧))
3534ralimdva 3172 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑥 < 𝑧) → (∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥 → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧))
3635impancom 455 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) → (𝑥 < 𝑧 → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧))
3736an32s 651 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) ∧ 𝑧 ∈ ℕ) → (𝑥 < 𝑧 → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧))
3837reximdva 3266 . . . 4 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) → (∃𝑧 ∈ ℕ 𝑥 < 𝑧 → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧))
392, 38mpd 15 . . 3 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)
40 fzfid 13345 . . . . 5 (𝜑 → (1...𝑀) ∈ Fin)
41 fvres 6680 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...𝑀) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = ((𝐺𝐾)‘𝑖))
4241adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = ((𝐺𝐾)‘𝑖))
43 elfznn 12940 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
44 fvco3 6751 . . . . . . . . . . . . . . 15 ((𝐾:ℕ⟶𝑇𝑖 ∈ ℕ) → ((𝐺𝐾)‘𝑖) = (𝐺‘(𝐾𝑖)))
4510, 43, 44syl2an 598 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐺𝐾)‘𝑖) = (𝐺‘(𝐾𝑖)))
4642, 45eqtrd 2859 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (𝐺‘(𝐾𝑖)))
4746adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (𝐺‘(𝐾𝑖)))
48 fvres 6680 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) = ((𝐺𝐾)‘𝑗))
4948ad2antll 728 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) = ((𝐺𝐾)‘𝑗))
50 elfznn 12940 . . . . . . . . . . . . . . 15 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
5150adantl 485 . . . . . . . . . . . . . 14 ((𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
52 fvco3 6751 . . . . . . . . . . . . . 14 ((𝐾:ℕ⟶𝑇𝑗 ∈ ℕ) → ((𝐺𝐾)‘𝑗) = (𝐺‘(𝐾𝑗)))
5310, 51, 52syl2an 598 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((𝐺𝐾)‘𝑗) = (𝐺‘(𝐾𝑗)))
5449, 53eqtrd 2859 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) = (𝐺‘(𝐾𝑗)))
5547, 54eqeq12d 2840 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) ↔ (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑗))))
56 2fveq3 6666 . . . . . . . . . . . 12 ((𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑗)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗)))))
5717a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑀) ⊆ ℕ)
58 elfznn 12940 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℕ)
5958ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑛 ∈ ℕ)
6059nnred 11649 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑛 ∈ ℝ)
61 ovolicc2.16 . . . . . . . . . . . . . . . . . . . . . . 23 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}
6261ssrab3 4043 . . . . . . . . . . . . . . . . . . . . . 22 𝑊 ⊆ ℕ
6362, 22sstri 3962 . . . . . . . . . . . . . . . . . . . . 21 𝑊 ⊆ ℝ
64 ovolicc2.17 . . . . . . . . . . . . . . . . . . . . . 22 𝑀 = inf(𝑊, ℝ, < )
6562, 5sseqtri 3989 . . . . . . . . . . . . . . . . . . . . . . 23 𝑊 ⊆ (ℤ‘1)
66 nnnfi 13338 . . . . . . . . . . . . . . . . . . . . . . . 24 ¬ ℕ ∈ Fin
67 ovolicc2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
6867elin2d 4161 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑈 ∈ Fin)
69 ssfi 8735 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑈 ∈ Fin ∧ 𝑇𝑈) → 𝑇 ∈ Fin)
7068, 12, 69sylancl 589 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑇 ∈ Fin)
7170adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑊 = ∅) → 𝑇 ∈ Fin)
7210adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑊 = ∅) → 𝐾:ℕ⟶𝑇)
73 2fveq3 6666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐾𝑖) = (𝐾𝑗) → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾𝑗))))
7473fveq2d 6665 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐾𝑖) = (𝐾𝑗) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗)))))
75 simpll 766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝜑)
76 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑖 ∈ ℕ)
77 ral0 4439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑚 ∈ ∅ 𝑛𝑚
78 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑊 = ∅)
7978raleqdv 3402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (∀𝑚𝑊 𝑛𝑚 ↔ ∀𝑚 ∈ ∅ 𝑛𝑚))
8077, 79mpbiri 261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ∀𝑚𝑊 𝑛𝑚)
8180ralrimivw 3178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ∀𝑛 ∈ ℕ ∀𝑚𝑊 𝑛𝑚)
82 rabid2 3372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (ℕ = {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ↔ ∀𝑛 ∈ ℕ ∀𝑚𝑊 𝑛𝑚)
8381, 82sylibr 237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ℕ = {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
8476, 83eleqtrd 2918 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
85 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑗 ∈ ℕ)
8685, 83eleqtrd 2918 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
87 ovolicc.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐴 ∈ ℝ)
88 ovolicc.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐵 ∈ ℝ)
89 ovolicc.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐴𝐵)
90 ovolicc2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
91 ovolicc2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
92 ovolicc2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
93 ovolicc2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
94 ovolicc2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
95 ovolicc2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐴𝐶)
9687, 88, 89, 90, 91, 67, 92, 4, 93, 11, 9, 94, 95, 8, 6, 61ovolicc2lem3 24130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗))))))
9775, 84, 86, 96syl12anc 835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗))))))
9874, 97syl5ibr 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝐾𝑖) = (𝐾𝑗) → 𝑖 = 𝑗))
9998ralrimivva 3186 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑊 = ∅) → ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℕ ((𝐾𝑖) = (𝐾𝑗) → 𝑖 = 𝑗))
100 dff13 7005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐾:ℕ–1-1𝑇 ↔ (𝐾:ℕ⟶𝑇 ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℕ ((𝐾𝑖) = (𝐾𝑗) → 𝑖 = 𝑗)))
10172, 99, 100sylanbrc 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑊 = ∅) → 𝐾:ℕ–1-1𝑇)
102 f1domg 8525 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑇 ∈ Fin → (𝐾:ℕ–1-1𝑇 → ℕ ≼ 𝑇))
10371, 101, 102sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑊 = ∅) → ℕ ≼ 𝑇)
104 domfi 8736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑇 ∈ Fin ∧ ℕ ≼ 𝑇) → ℕ ∈ Fin)
10570, 103, 104syl2an2r 684 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑊 = ∅) → ℕ ∈ Fin)
106105ex 416 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑊 = ∅ → ℕ ∈ Fin))
107106necon3bd 3028 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (¬ ℕ ∈ Fin → 𝑊 ≠ ∅))
10866, 107mpi 20 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑊 ≠ ∅)
109 infssuzcl 12329 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑊 ⊆ (ℤ‘1) ∧ 𝑊 ≠ ∅) → inf(𝑊, ℝ, < ) ∈ 𝑊)
11065, 108, 109sylancr 590 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → inf(𝑊, ℝ, < ) ∈ 𝑊)
11164, 110eqeltrid 2920 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀𝑊)
11263, 111sseldi 3951 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℝ)
113112ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑀 ∈ ℝ)
11463sseli 3949 . . . . . . . . . . . . . . . . . . . 20 (𝑚𝑊𝑚 ∈ ℝ)
115114adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑚 ∈ ℝ)
116 elfzle2 12915 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑀) → 𝑛𝑀)
117116ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑛𝑀)
118 infssuzle 12328 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑊 ⊆ (ℤ‘1) ∧ 𝑚𝑊) → inf(𝑊, ℝ, < ) ≤ 𝑚)
11965, 118mpan 689 . . . . . . . . . . . . . . . . . . . . 21 (𝑚𝑊 → inf(𝑊, ℝ, < ) ≤ 𝑚)
12064, 119eqbrtrid 5087 . . . . . . . . . . . . . . . . . . . 20 (𝑚𝑊𝑀𝑚)
121120adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑀𝑚)
12260, 113, 115, 117, 121letrd 10795 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑛𝑚)
123122ralrimiva 3177 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑚𝑊 𝑛𝑚)
12457, 123ssrabdv 4036 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑀) ⊆ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
125124adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (1...𝑀) ⊆ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
126 simprl 770 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑖 ∈ (1...𝑀))
127125, 126sseldd 3954 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
128 simprr 772 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑗 ∈ (1...𝑀))
129125, 128sseldd 3954 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
130127, 129jca 515 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚}))
131130, 96syldan 594 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗))))))
13256, 131syl5ibr 249 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑗)) → 𝑖 = 𝑗))
13355, 132sylbid 243 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) → 𝑖 = 𝑗))
134133ralrimivva 3186 . . . . . . . . 9 (𝜑 → ∀𝑖 ∈ (1...𝑀)∀𝑗 ∈ (1...𝑀)((((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) → 𝑖 = 𝑗))
135 dff13 7005 . . . . . . . . 9 (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1→ℕ ↔ (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ ∧ ∀𝑖 ∈ (1...𝑀)∀𝑗 ∈ (1...𝑀)((((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) → 𝑖 = 𝑗)))
13619, 134, 135sylanbrc 586 . . . . . . . 8 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1→ℕ)
137 f1f1orn 6617 . . . . . . . 8 (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1→ℕ → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺𝐾) ↾ (1...𝑀)))
138136, 137syl 17 . . . . . . 7 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺𝐾) ↾ (1...𝑀)))
139 f1oeq3 6597 . . . . . . . 8 (((𝐺𝐾) “ (1...𝑀)) = ran ((𝐺𝐾) ↾ (1...𝑀)) → (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺𝐾) “ (1...𝑀)) ↔ ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺𝐾) ↾ (1...𝑀))))
1403, 139ax-mp 5 . . . . . . 7 (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺𝐾) “ (1...𝑀)) ↔ ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺𝐾) ↾ (1...𝑀)))
141138, 140sylibr 237 . . . . . 6 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺𝐾) “ (1...𝑀)))
142 f1ofo 6613 . . . . . 6 (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺𝐾) “ (1...𝑀)) → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺𝐾) “ (1...𝑀)))
143141, 142syl 17 . . . . 5 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺𝐾) “ (1...𝑀)))
144 fofi 8807 . . . . 5 (((1...𝑀) ∈ Fin ∧ ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺𝐾) “ (1...𝑀))) → ((𝐺𝐾) “ (1...𝑀)) ∈ Fin)
14540, 143, 144syl2anc 587 . . . 4 (𝜑 → ((𝐺𝐾) “ (1...𝑀)) ∈ Fin)
146 fimaxre2 11583 . . . 4 ((((𝐺𝐾) “ (1...𝑀)) ⊆ ℝ ∧ ((𝐺𝐾) “ (1...𝑀)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥)
14723, 145, 146syl2anc 587 . . 3 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥)
14839, 147r19.29a 3281 . 2 (𝜑 → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)
14988, 87resubcld 11066 . . . . 5 (𝜑 → (𝐵𝐴) ∈ ℝ)
150149rexrd 10689 . . . 4 (𝜑 → (𝐵𝐴) ∈ ℝ*)
151150adantr 484 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ∈ ℝ*)
152 fzfid 13345 . . . . . 6 (𝜑 → (1...𝑧) ∈ Fin)
153 elfznn 12940 . . . . . . . . 9 (𝑗 ∈ (1...𝑧) → 𝑗 ∈ ℕ)
154 eqid 2824 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
155154ovolfsf 24082 . . . . . . . . . . 11 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
15691, 155syl 17 . . . . . . . . . 10 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
157156ffvelrnda 6842 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞))
158153, 157sylan2 595 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞))
159 elrege0 12841 . . . . . . . 8 ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗)))
160158, 159sylib 221 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑧)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗)))
161160simpld 498 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
162152, 161fsumrecl 15091 . . . . 5 (𝜑 → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
163162adantr 484 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
164163rexrd 10689 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ*)
165154, 90ovolsf 24083 . . . . . . . . 9 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
16691, 165syl 17 . . . . . . . 8 (𝜑𝑆:ℕ⟶(0[,)+∞))
167166frnd 6510 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
168 rge0ssre 12843 . . . . . . 7 (0[,)+∞) ⊆ ℝ
169167, 168sstrdi 3965 . . . . . 6 (𝜑 → ran 𝑆 ⊆ ℝ)
170 ressxr 10683 . . . . . 6 ℝ ⊆ ℝ*
171169, 170sstrdi 3965 . . . . 5 (𝜑 → ran 𝑆 ⊆ ℝ*)
172 supxrcl 12705 . . . . 5 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
173171, 172syl 17 . . . 4 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
174173adantr 484 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
175149adantr 484 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ∈ ℝ)
17621sselda 3953 . . . . . . 7 ((𝜑𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑗 ∈ ℕ)
177168, 157sseldi 3951 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
178176, 177syldan 594 . . . . . 6 ((𝜑𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
179145, 178fsumrecl 15091 . . . . 5 (𝜑 → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
180179adantr 484 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
181 inss2 4191 . . . . . . . . . . 11 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
182 fss 6517 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
18391, 181, 182sylancl 589 . . . . . . . . . 10 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
18462, 111sseldi 3951 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℕ)
18514, 184ffvelrnd 6843 . . . . . . . . . . 11 (𝜑 → (𝐾𝑀) ∈ 𝑈)
1864, 185ffvelrnd 6843 . . . . . . . . . 10 (𝜑 → (𝐺‘(𝐾𝑀)) ∈ ℕ)
187183, 186ffvelrnd 6843 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐺‘(𝐾𝑀))) ∈ (ℝ × ℝ))
188 xp2nd 7717 . . . . . . . . 9 ((𝐹‘(𝐺‘(𝐾𝑀))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) ∈ ℝ)
189187, 188syl 17 . . . . . . . 8 (𝜑 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) ∈ ℝ)
19012, 8sseldi 3951 . . . . . . . . . . 11 (𝜑𝐶𝑈)
1914, 190ffvelrnd 6843 . . . . . . . . . 10 (𝜑 → (𝐺𝐶) ∈ ℕ)
192183, 191ffvelrnd 6843 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐺𝐶)) ∈ (ℝ × ℝ))
193 xp1st 7716 . . . . . . . . 9 ((𝐹‘(𝐺𝐶)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝐶))) ∈ ℝ)
194192, 193syl 17 . . . . . . . 8 (𝜑 → (1st ‘(𝐹‘(𝐺𝐶))) ∈ ℝ)
195189, 194resubcld 11066 . . . . . . 7 (𝜑 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) ∈ ℝ)
196 fveq2 6661 . . . . . . . . . 10 (𝑗 = (𝐺‘(𝐾𝑖)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) = (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))))
197177recnd 10667 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
198176, 197syldan 594 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
199196, 40, 141, 46, 198fsumf1o 15080 . . . . . . . . 9 (𝜑 → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) = Σ𝑖 ∈ (1...𝑀)(((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))))
2004adantr 484 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐺:𝑈⟶ℕ)
201 ffvelrn 6840 . . . . . . . . . . . . 13 ((𝐾:ℕ⟶𝑈𝑖 ∈ ℕ) → (𝐾𝑖) ∈ 𝑈)
20214, 43, 201syl2an 598 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝑈)
203200, 202ffvelrnd 6843 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘(𝐾𝑖)) ∈ ℕ)
204154ovolfsval 24081 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺‘(𝐾𝑖)) ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
20591, 203, 204syl2an2r 684 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
206205sumeq2dv 15060 . . . . . . . . 9 (𝜑 → Σ𝑖 ∈ (1...𝑀)(((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))) = Σ𝑖 ∈ (1...𝑀)((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
207183adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
2084adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ ℕ) → 𝐺:𝑈⟶ℕ)
20914ffvelrnda 6842 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ ℕ) → (𝐾𝑖) ∈ 𝑈)
210208, 209ffvelrnd 6843 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ℕ) → (𝐺‘(𝐾𝑖)) ∈ ℕ)
211207, 210ffvelrnd 6843 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾𝑖))) ∈ (ℝ × ℝ))
212 xp2nd 7717 . . . . . . . . . . . . . 14 ((𝐹‘(𝐺‘(𝐾𝑖))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
213211, 212syl 17 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ℕ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
21443, 213sylan2 595 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
215214recnd 10667 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℂ)
216183adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐹:ℕ⟶(ℝ × ℝ))
217216, 203ffvelrnd 6843 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐹‘(𝐺‘(𝐾𝑖))) ∈ (ℝ × ℝ))
218 xp1st 7716 . . . . . . . . . . . . 13 ((𝐹‘(𝐺‘(𝐾𝑖))) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
219217, 218syl 17 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
220219recnd 10667 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℂ)
22140, 215, 220fsumsub 15143 . . . . . . . . . 10 (𝜑 → Σ𝑖 ∈ (1...𝑀)((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))) = (Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
222 fzfid 13345 . . . . . . . . . . . . . 14 (𝜑 → (1...(𝑀 − 1)) ∈ Fin)
223 elfznn 12940 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℕ)
224223, 213sylan2 595 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
225222, 224fsumrecl 15091 . . . . . . . . . . . . 13 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
226225recnd 10667 . . . . . . . . . . . 12 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℂ)
227189recnd 10667 . . . . . . . . . . . 12 (𝜑 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) ∈ ℂ)
22865, 111sseldi 3951 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (ℤ‘1))
229 2fveq3 6666 . . . . . . . . . . . . . . 15 (𝑖 = 𝑀 → (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑀)))
230229fveq2d 6665 . . . . . . . . . . . . . 14 (𝑖 = 𝑀 → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾𝑀))))
231230fveq2d 6665 . . . . . . . . . . . . 13 (𝑖 = 𝑀 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))))
232228, 215, 231fsumm1 15106 . . . . . . . . . . . 12 (𝜑 → Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) + (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀))))))
233226, 227, 232comraddd 10852 . . . . . . . . . . 11 (𝜑 → Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
234 2fveq3 6666 . . . . . . . . . . . . . . 15 (𝑖 = 1 → (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾‘1)))
235234fveq2d 6665 . . . . . . . . . . . . . 14 (𝑖 = 1 → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾‘1))))
236235fveq2d 6665 . . . . . . . . . . . . 13 (𝑖 = 1 → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘1)))))
237228, 220, 236fsum1p 15108 . . . . . . . . . . . 12 (𝜑 → Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = ((1st ‘(𝐹‘(𝐺‘(𝐾‘1)))) + Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
2385, 6, 7, 8algr0 15914 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐾‘1) = 𝐶)
239238fveq2d 6665 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺‘(𝐾‘1)) = (𝐺𝐶))
240239fveq2d 6665 . . . . . . . . . . . . . 14 (𝜑 → (𝐹‘(𝐺‘(𝐾‘1))) = (𝐹‘(𝐺𝐶)))
241240fveq2d 6665 . . . . . . . . . . . . 13 (𝜑 → (1st ‘(𝐹‘(𝐺‘(𝐾‘1)))) = (1st ‘(𝐹‘(𝐺𝐶))))
2427peano2zd 12087 . . . . . . . . . . . . . . 15 (𝜑 → (1 + 1) ∈ ℤ)
243184nnzd 12083 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℤ)
244 1z 12009 . . . . . . . . . . . . . . . . . 18 1 ∈ ℤ
245 fzp1ss 12962 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℤ → ((1 + 1)...𝑀) ⊆ (1...𝑀))
246244, 245mp1i 13 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1 + 1)...𝑀) ⊆ (1...𝑀))
247246sselda 3953 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ ((1 + 1)...𝑀)) → 𝑖 ∈ (1...𝑀))
248247, 220syldan 594 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ((1 + 1)...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℂ)
249 2fveq3 6666 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑗 + 1) → (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾‘(𝑗 + 1))))
250249fveq2d 6665 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑗 + 1) → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))))
251250fveq2d 6665 . . . . . . . . . . . . . . 15 (𝑖 = (𝑗 + 1) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))))
2527, 242, 243, 248, 251fsumshftm 15136 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))))
253 ax-1cn 10593 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
254253, 253pncan3oi 10900 . . . . . . . . . . . . . . . . 17 ((1 + 1) − 1) = 1
255254oveq1i 7159 . . . . . . . . . . . . . . . 16 (((1 + 1) − 1)...(𝑀 − 1)) = (1...(𝑀 − 1))
256255sumeq1i 15055 . . . . . . . . . . . . . . 15 Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑗 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))))
257 fvoveq1 7172 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → (𝐾‘(𝑗 + 1)) = (𝐾‘(𝑖 + 1)))
258257fveq2d 6665 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → (𝐺‘(𝐾‘(𝑗 + 1))) = (𝐺‘(𝐾‘(𝑖 + 1))))
259258fveq2d 6665 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → (𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))) = (𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
260259fveq2d 6665 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))
261260cbvsumv 15053 . . . . . . . . . . . . . . 15 Σ𝑗 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
262256, 261eqtri 2847 . . . . . . . . . . . . . 14 Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
263252, 262syl6eq 2875 . . . . . . . . . . . . 13 (𝜑 → Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))
264241, 263oveq12d 7167 . . . . . . . . . . . 12 (𝜑 → ((1st ‘(𝐹‘(𝐺‘(𝐾‘1)))) + Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))) = ((1st ‘(𝐹‘(𝐺𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
265237, 264eqtrd 2859 . . . . . . . . . . 11 (𝜑 → Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = ((1st ‘(𝐹‘(𝐺𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
266233, 265oveq12d 7167 . . . . . . . . . 10 (𝜑 → (Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))) = (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖))))) − ((1st ‘(𝐹‘(𝐺𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
267194recnd 10667 . . . . . . . . . . 11 (𝜑 → (1st ‘(𝐹‘(𝐺𝐶))) ∈ ℂ)
268 peano2nn 11646 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ ℕ → (𝑖 + 1) ∈ ℕ)
269 ffvelrn 6840 . . . . . . . . . . . . . . . . . 18 ((𝐾:ℕ⟶𝑈 ∧ (𝑖 + 1) ∈ ℕ) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
27014, 268, 269syl2an 598 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ ℕ) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
271208, 270ffvelrnd 6843 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ ℕ) → (𝐺‘(𝐾‘(𝑖 + 1))) ∈ ℕ)
272207, 271ffvelrnd 6843 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))) ∈ (ℝ × ℝ))
273 xp1st 7716 . . . . . . . . . . . . . . 15 ((𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
274272, 273syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ ℕ) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
275223, 274sylan2 595 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
276222, 275fsumrecl 15091 . . . . . . . . . . . 12 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
277276recnd 10667 . . . . . . . . . . 11 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℂ)
278227, 226, 267, 277addsub4d 11042 . . . . . . . . . 10 (𝜑 → (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖))))) − ((1st ‘(𝐹‘(𝐺𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))) = (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
279221, 266, 2783eqtrd 2863 . . . . . . . . 9 (𝜑 → Σ𝑖 ∈ (1...𝑀)((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))) = (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
280199, 206, 2793eqtrd 2863 . . . . . . . 8 (𝜑 → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) = (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
281280, 179eqeltrrd 2917 . . . . . . 7 (𝜑 → (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))) ∈ ℝ)
282 fveq2 6661 . . . . . . . . . . . . . . 15 (𝑛 = 𝑀 → (𝐾𝑛) = (𝐾𝑀))
283282eleq2d 2901 . . . . . . . . . . . . . 14 (𝑛 = 𝑀 → (𝐵 ∈ (𝐾𝑛) ↔ 𝐵 ∈ (𝐾𝑀)))
284283, 61elrab2 3669 . . . . . . . . . . . . 13 (𝑀𝑊 ↔ (𝑀 ∈ ℕ ∧ 𝐵 ∈ (𝐾𝑀)))
285111, 284sylib 221 . . . . . . . . . . . 12 (𝜑 → (𝑀 ∈ ℕ ∧ 𝐵 ∈ (𝐾𝑀)))
286285simprd 499 . . . . . . . . . . 11 (𝜑𝐵 ∈ (𝐾𝑀))
28787, 88, 89, 90, 91, 67, 92, 4, 93ovolicc2lem1 24128 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐾𝑀) ∈ 𝑈) → (𝐵 ∈ (𝐾𝑀) ↔ (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑀)))) < 𝐵𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))))))
288185, 287mpdan 686 . . . . . . . . . . 11 (𝜑 → (𝐵 ∈ (𝐾𝑀) ↔ (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑀)))) < 𝐵𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))))))
289286, 288mpbid 235 . . . . . . . . . 10 (𝜑 → (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑀)))) < 𝐵𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀))))))
290289simp3d 1141 . . . . . . . . 9 (𝜑𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))))
29187, 88, 89, 90, 91, 67, 92, 4, 93ovolicc2lem1 24128 . . . . . . . . . . . 12 ((𝜑𝐶𝑈) → (𝐴𝐶 ↔ (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝐶))) < 𝐴𝐴 < (2nd ‘(𝐹‘(𝐺𝐶))))))
292190, 291mpdan 686 . . . . . . . . . . 11 (𝜑 → (𝐴𝐶 ↔ (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝐶))) < 𝐴𝐴 < (2nd ‘(𝐹‘(𝐺𝐶))))))
29395, 292mpbid 235 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝐶))) < 𝐴𝐴 < (2nd ‘(𝐹‘(𝐺𝐶)))))
294293simp2d 1140 . . . . . . . . 9 (𝜑 → (1st ‘(𝐹‘(𝐺𝐶))) < 𝐴)
29588, 194, 189, 87, 290, 294lt2subd 11262 . . . . . . . 8 (𝜑 → (𝐵𝐴) < ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))))
296149, 195, 295ltled 10786 . . . . . . 7 (𝜑 → (𝐵𝐴) ≤ ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))))
297223adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℕ)
298 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (1...(𝑀 − 1)))
299243adantr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℤ)
300 elfzm11 12982 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 ∈ (1...(𝑀 − 1)) ↔ (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖𝑖 < 𝑀)))
301244, 299, 300sylancr 590 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (1...(𝑀 − 1)) ↔ (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖𝑖 < 𝑀)))
302298, 301mpbid 235 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖𝑖 < 𝑀))
303302simp3d 1141 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 < 𝑀)
304297nnred 11649 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℝ)
305112adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℝ)
306304, 305ltnled 10785 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 < 𝑀 ↔ ¬ 𝑀𝑖))
307303, 306mpbid 235 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → ¬ 𝑀𝑖)
308 infssuzle 12328 . . . . . . . . . . . . . . . . . . . . 21 ((𝑊 ⊆ (ℤ‘1) ∧ 𝑖𝑊) → inf(𝑊, ℝ, < ) ≤ 𝑖)
30965, 308mpan 689 . . . . . . . . . . . . . . . . . . . 20 (𝑖𝑊 → inf(𝑊, ℝ, < ) ≤ 𝑖)
31064, 309eqbrtrid 5087 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑊𝑀𝑖)
311307, 310nsyl 142 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → ¬ 𝑖𝑊)
312297, 311jca 515 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ ℕ ∧ ¬ 𝑖𝑊))
31387, 88, 89, 90, 91, 67, 92, 4, 93, 11, 9, 94, 95, 8, 6, 61ovolicc2lem2 24129 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ¬ 𝑖𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵)
314312, 313syldan 594 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵)
315314iftrued 4458 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))), 𝐵) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
316 2fveq3 6666 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝐾𝑖) → (𝐹‘(𝐺𝑡)) = (𝐹‘(𝐺‘(𝐾𝑖))))
317316fveq2d 6665 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (𝐾𝑖) → (2nd ‘(𝐹‘(𝐺𝑡))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
318317breq1d 5062 . . . . . . . . . . . . . . . . . 18 (𝑡 = (𝐾𝑖) → ((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵))
319318, 317ifbieq1d 4473 . . . . . . . . . . . . . . . . 17 (𝑡 = (𝐾𝑖) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))), 𝐵))
320 fveq2 6661 . . . . . . . . . . . . . . . . 17 (𝑡 = (𝐾𝑖) → (𝐻𝑡) = (𝐻‘(𝐾𝑖)))
321319, 320eleq12d 2910 . . . . . . . . . . . . . . . 16 (𝑡 = (𝐾𝑖) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡) ↔ if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))), 𝐵) ∈ (𝐻‘(𝐾𝑖))))
32294ralrimiva 3177 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
323322adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
324 ffvelrn 6840 . . . . . . . . . . . . . . . . 17 ((𝐾:ℕ⟶𝑇𝑖 ∈ ℕ) → (𝐾𝑖) ∈ 𝑇)
32510, 223, 324syl2an 598 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝐾𝑖) ∈ 𝑇)
326321, 323, 325rspcdva 3611 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))), 𝐵) ∈ (𝐻‘(𝐾𝑖)))
327315, 326eqeltrrd 2917 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ (𝐻‘(𝐾𝑖)))
3285, 6, 7, 8, 9algrp1 15916 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ℕ) → (𝐾‘(𝑖 + 1)) = (𝐻‘(𝐾𝑖)))
329223, 328sylan2 595 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝐾‘(𝑖 + 1)) = (𝐻‘(𝐾𝑖)))
330327, 329eleqtrrd 2919 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ (𝐾‘(𝑖 + 1)))
331223, 270sylan2 595 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
33287, 88, 89, 90, 91, 67, 92, 4, 93ovolicc2lem1 24128 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐾‘(𝑖 + 1)) ∈ 𝑈) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ (𝐾‘(𝑖 + 1)) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
333331, 332syldan 594 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ (𝐾‘(𝑖 + 1)) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
334330, 333mpbid 235 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
335334simp2d 1140 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
336275, 224, 335ltled 10786 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ≤ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
337222, 275, 224, 336fsumle 15154 . . . . . . . . 9 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ≤ Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
338225, 276subge0d 11228 . . . . . . . . 9 (𝜑 → (0 ≤ (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))) ↔ Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ≤ Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
339337, 338mpbird 260 . . . . . . . 8 (𝜑 → 0 ≤ (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
340225, 276resubcld 11066 . . . . . . . . 9 (𝜑 → (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))) ∈ ℝ)
341195, 340addge01d 11226 . . . . . . . 8 (𝜑 → (0 ≤ (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) ≤ (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))))
342339, 341mpbid 235 . . . . . . 7 (𝜑 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) ≤ (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
343149, 195, 281, 296, 342letrd 10795 . . . . . 6 (𝜑 → (𝐵𝐴) ≤ (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
344343, 280breqtrrd 5080 . . . . 5 (𝜑 → (𝐵𝐴) ≤ Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗))
345344adantr 484 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ≤ Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗))
346 fzfid 13345 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (1...𝑧) ∈ Fin)
347161adantlr 714 . . . . 5 (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
348160simprd 499 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑧)) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗))
349348adantlr 714 . . . . 5 (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗))
35021adantr 484 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℕ) → ((𝐺𝐾) “ (1...𝑀)) ⊆ ℕ)
351350sselda 3953 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ ℕ)
352351nnred 11649 . . . . . . . . . 10 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ ℝ)
35328ad2antlr 726 . . . . . . . . . 10 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑧 ∈ ℝ)
354 ltle 10727 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 < 𝑧𝑦𝑧))
355352, 353, 354syl2anc 587 . . . . . . . . 9 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → (𝑦 < 𝑧𝑦𝑧))
356351, 5eleqtrdi 2926 . . . . . . . . . 10 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ (ℤ‘1))
357 nnz 12001 . . . . . . . . . . 11 (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
358357ad2antlr 726 . . . . . . . . . 10 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑧 ∈ ℤ)
359 elfz5 12903 . . . . . . . . . 10 ((𝑦 ∈ (ℤ‘1) ∧ 𝑧 ∈ ℤ) → (𝑦 ∈ (1...𝑧) ↔ 𝑦𝑧))
360356, 358, 359syl2anc 587 . . . . . . . . 9 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → (𝑦 ∈ (1...𝑧) ↔ 𝑦𝑧))
361355, 360sylibrd 262 . . . . . . . 8 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → (𝑦 < 𝑧𝑦 ∈ (1...𝑧)))
362361ralimdva 3172 . . . . . . 7 ((𝜑𝑧 ∈ ℕ) → (∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧 → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧)))
363362impr 458 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧))
364 dfss3 3941 . . . . . 6 (((𝐺𝐾) “ (1...𝑀)) ⊆ (1...𝑧) ↔ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧))
365363, 364sylibr 237 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → ((𝐺𝐾) “ (1...𝑀)) ⊆ (1...𝑧))
366346, 347, 349, 365fsumless 15151 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ≤ Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗))
367175, 180, 163, 345, 366letrd 10795 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ≤ Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗))
368 eqidd 2825 . . . . . 6 (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) = (((abs ∘ − ) ∘ 𝐹)‘𝑗))
369 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → 𝑧 ∈ ℕ)
370369, 5eleqtrdi 2926 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → 𝑧 ∈ (ℤ‘1))
371347recnd 10667 . . . . . 6 (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
372368, 370, 371fsumser 15087 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑧))
37390fveq1i 6662 . . . . 5 (𝑆𝑧) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑧)
374372, 373eqtr4di 2877 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) = (𝑆𝑧))
375166ffnd 6504 . . . . . 6 (𝜑𝑆 Fn ℕ)
376 fnfvelrn 6839 . . . . . 6 ((𝑆 Fn ℕ ∧ 𝑧 ∈ ℕ) → (𝑆𝑧) ∈ ran 𝑆)
377375, 369, 376syl2an2r 684 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝑆𝑧) ∈ ran 𝑆)
378 supxrub 12714 . . . . 5 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑧) ∈ ran 𝑆) → (𝑆𝑧) ≤ sup(ran 𝑆, ℝ*, < ))
379171, 377, 378syl2an2r 684 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝑆𝑧) ≤ sup(ran 𝑆, ℝ*, < ))
380374, 379eqbrtrd 5074 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
381151, 164, 174, 367, 380xrletrd 12552 . 2 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
382148, 381rexlimddv 3283 1 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  {crab 3137  cin 3918  wss 3919  c0 4276  ifcif 4450  𝒫 cpw 4522  {csn 4550   cuni 4824   class class class wbr 5052   × cxp 5540  ran crn 5543  cres 5544  cima 5545  ccom 5546   Fn wfn 6338  wf 6339  1-1wf1 6340  ontowfo 6341  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  cdom 8503  Fincfn 8505  supcsup 8901  infcinf 8902  cc 10533  cr 10534  0cc0 10535  1c1 10536   + caddc 10538  +∞cpnf 10670  *cxr 10672   < clt 10673  cle 10674  cmin 10868  cn 11634  cz 11978  cuz 12240  (,)cioo 12735  [,)cico 12737  [,]cicc 12738  ...cfz 12894  seqcseq 13373  abscabs 14593  Σcsu 15042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-ioo 12739  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043
This theorem is referenced by:  ovolicc2lem5  24132
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