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Theorem ovolicc2lem4 23507
Description: Lemma for ovolicc2 23509. (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 11490 . . . . 5 (𝑥 ∈ ℝ → ∃𝑧 ∈ ℕ 𝑥 < 𝑧)
21ad2antlr 698 . . . 4 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) → ∃𝑧 ∈ ℕ 𝑥 < 𝑧)
3 df-ima 5262 . . . . . . . . . . . . . . . 16 ((𝐺𝐾) “ (1...𝑀)) = ran ((𝐺𝐾) ↾ (1...𝑀))
4 ovolicc2.8 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺:𝑈⟶ℕ)
5 nnuz 11924 . . . . . . . . . . . . . . . . . . . . 21 ℕ = (ℤ‘1)
6 ovolicc2.15 . . . . . . . . . . . . . . . . . . . . 21 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
7 1zzd 11609 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ∈ ℤ)
8 ovolicc2.14 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐶𝑇)
9 ovolicc2.11 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐻:𝑇𝑇)
105, 6, 7, 8, 9algrf 15493 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐾:ℕ⟶𝑇)
11 ovolicc2.10 . . . . . . . . . . . . . . . . . . . . 21 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
12 ssrab2 3834 . . . . . . . . . . . . . . . . . . . . 21 {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} ⊆ 𝑈
1311, 12eqsstri 3782 . . . . . . . . . . . . . . . . . . . 20 𝑇𝑈
14 fss 6196 . . . . . . . . . . . . . . . . . . . 20 ((𝐾:ℕ⟶𝑇𝑇𝑈) → 𝐾:ℕ⟶𝑈)
1510, 13, 14sylancl 566 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐾:ℕ⟶𝑈)
16 fco 6198 . . . . . . . . . . . . . . . . . . 19 ((𝐺:𝑈⟶ℕ ∧ 𝐾:ℕ⟶𝑈) → (𝐺𝐾):ℕ⟶ℕ)
174, 15, 16syl2anc 565 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐺𝐾):ℕ⟶ℕ)
18 elfznn 12576 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
1918ssriv 3754 . . . . . . . . . . . . . . . . . 18 (1...𝑀) ⊆ ℕ
20 fssres 6210 . . . . . . . . . . . . . . . . . 18 (((𝐺𝐾):ℕ⟶ℕ ∧ (1...𝑀) ⊆ ℕ) → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ)
2117, 19, 20sylancl 566 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ)
22 frn 6193 . . . . . . . . . . . . . . . . 17 (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ → ran ((𝐺𝐾) ↾ (1...𝑀)) ⊆ ℕ)
2321, 22syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ran ((𝐺𝐾) ↾ (1...𝑀)) ⊆ ℕ)
243, 23syl5eqss 3796 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐺𝐾) “ (1...𝑀)) ⊆ ℕ)
25 nnssre 11225 . . . . . . . . . . . . . . 15 ℕ ⊆ ℝ
2624, 25syl6ss 3762 . . . . . . . . . . . . . 14 (𝜑 → ((𝐺𝐾) “ (1...𝑀)) ⊆ ℝ)
2726ad3antrrr 701 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → ((𝐺𝐾) “ (1...𝑀)) ⊆ ℝ)
28 simpr 471 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀)))
2927, 28sseldd 3751 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ ℝ)
30 simpllr 752 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑥 ∈ ℝ)
31 nnre 11228 . . . . . . . . . . . . 13 (𝑧 ∈ ℕ → 𝑧 ∈ ℝ)
3231ad2antlr 698 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑧 ∈ ℝ)
33 lelttr 10329 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦𝑥𝑥 < 𝑧) → 𝑦 < 𝑧))
3429, 30, 32, 33syl3anc 1475 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → ((𝑦𝑥𝑥 < 𝑧) → 𝑦 < 𝑧))
3534ancomsd 456 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → ((𝑥 < 𝑧𝑦𝑥) → 𝑦 < 𝑧))
3635expdimp 440 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) ∧ 𝑥 < 𝑧) → (𝑦𝑥𝑦 < 𝑧))
3736an32s 623 . . . . . . . 8 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑥 < 𝑧) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → (𝑦𝑥𝑦 < 𝑧))
3837ralimdva 3110 . . . . . . 7 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑥 < 𝑧) → (∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥 → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧))
3938impancom 439 . . . . . 6 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) → (𝑥 < 𝑧 → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧))
4039an32s 623 . . . . 5 ((((𝜑𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) ∧ 𝑧 ∈ ℕ) → (𝑥 < 𝑧 → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧))
4140reximdva 3164 . . . 4 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) → (∃𝑧 ∈ ℕ 𝑥 < 𝑧 → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧))
422, 41mpd 15 . . 3 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥) → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)
43 fzfid 12979 . . . . 5 (𝜑 → (1...𝑀) ∈ Fin)
44 fvres 6348 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...𝑀) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = ((𝐺𝐾)‘𝑖))
4544adantl 467 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = ((𝐺𝐾)‘𝑖))
46 fvco3 6417 . . . . . . . . . . . . . . 15 ((𝐾:ℕ⟶𝑇𝑖 ∈ ℕ) → ((𝐺𝐾)‘𝑖) = (𝐺‘(𝐾𝑖)))
4710, 18, 46syl2an 575 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐺𝐾)‘𝑖) = (𝐺‘(𝐾𝑖)))
4845, 47eqtrd 2804 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (𝐺‘(𝐾𝑖)))
4948adantrr 688 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (𝐺‘(𝐾𝑖)))
50 fvres 6348 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) = ((𝐺𝐾)‘𝑗))
5150ad2antll 700 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) = ((𝐺𝐾)‘𝑗))
52 elfznn 12576 . . . . . . . . . . . . . . 15 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
5352adantl 467 . . . . . . . . . . . . . 14 ((𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
54 fvco3 6417 . . . . . . . . . . . . . 14 ((𝐾:ℕ⟶𝑇𝑗 ∈ ℕ) → ((𝐺𝐾)‘𝑗) = (𝐺‘(𝐾𝑗)))
5510, 53, 54syl2an 575 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((𝐺𝐾)‘𝑗) = (𝐺‘(𝐾𝑗)))
5651, 55eqtrd 2804 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) = (𝐺‘(𝐾𝑗)))
5749, 56eqeq12d 2785 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) ↔ (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑗))))
58 fveq2 6332 . . . . . . . . . . . . 13 ((𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑗)) → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾𝑗))))
5958fveq2d 6336 . . . . . . . . . . . 12 ((𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑗)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗)))))
6019a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑀) ⊆ ℕ)
61 elfznn 12576 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℕ)
6261ad2antlr 698 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑛 ∈ ℕ)
6362nnred 11236 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑛 ∈ ℝ)
64 ovolicc2.16 . . . . . . . . . . . . . . . . . . . . . . 23 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}
65 ssrab2 3834 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)} ⊆ ℕ
6664, 65eqsstri 3782 . . . . . . . . . . . . . . . . . . . . . 22 𝑊 ⊆ ℕ
6766, 25sstri 3759 . . . . . . . . . . . . . . . . . . . . 21 𝑊 ⊆ ℝ
68 ovolicc2.17 . . . . . . . . . . . . . . . . . . . . . 22 𝑀 = inf(𝑊, ℝ, < )
6966, 5sseqtri 3784 . . . . . . . . . . . . . . . . . . . . . . 23 𝑊 ⊆ (ℤ‘1)
70 1z 11608 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ ℤ
715uzinf 12971 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
7270, 71ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 ¬ ℕ ∈ Fin
73 ovolicc2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
74 elin 3945 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin) ↔ (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin))
7573, 74sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin))
7675simprd 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑈 ∈ Fin)
77 ssfi 8335 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑈 ∈ Fin ∧ 𝑇𝑈) → 𝑇 ∈ Fin)
7876, 13, 77sylancl 566 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑇 ∈ Fin)
7978adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑊 = ∅) → 𝑇 ∈ Fin)
8010adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑊 = ∅) → 𝐾:ℕ⟶𝑇)
81 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐾𝑖) = (𝐾𝑗) → (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑗)))
8281fveq2d 6336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐾𝑖) = (𝐾𝑗) → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾𝑗))))
8382fveq2d 6336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐾𝑖) = (𝐾𝑗) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗)))))
84 simpll 742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝜑)
85 simprl 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑖 ∈ ℕ)
86 ral0 4215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑚 ∈ ∅ 𝑛𝑚
87 simplr 744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑊 = ∅)
8887raleqdv 3292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (∀𝑚𝑊 𝑛𝑚 ↔ ∀𝑚 ∈ ∅ 𝑛𝑚))
8986, 88mpbiri 248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ∀𝑚𝑊 𝑛𝑚)
9089ralrimivw 3115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ∀𝑛 ∈ ℕ ∀𝑚𝑊 𝑛𝑚)
91 rabid2 3266 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (ℕ = {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ↔ ∀𝑛 ∈ ℕ ∀𝑚𝑊 𝑛𝑚)
9290, 91sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ℕ = {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
9385, 92eleqtrd 2851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
94 simprr 748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑗 ∈ ℕ)
9594, 92eleqtrd 2851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
96 ovolicc.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐴 ∈ ℝ)
97 ovolicc.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐵 ∈ ℝ)
98 ovolicc.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐴𝐵)
99 ovolicc2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
100 ovolicc2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
101 ovolicc2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
102 ovolicc2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
103 ovolicc2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
104 ovolicc2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝐴𝐶)
10596, 97, 98, 99, 100, 73, 101, 4, 102, 11, 9, 103, 104, 8, 6, 64ovolicc2lem3 23506 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗))))))
10684, 93, 95, 105syl12anc 1473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗))))))
10783, 106syl5ibr 236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝐾𝑖) = (𝐾𝑗) → 𝑖 = 𝑗))
108107ralrimivva 3119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑊 = ∅) → ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℕ ((𝐾𝑖) = (𝐾𝑗) → 𝑖 = 𝑗))
109 dff13 6654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐾:ℕ–1-1𝑇 ↔ (𝐾:ℕ⟶𝑇 ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℕ ((𝐾𝑖) = (𝐾𝑗) → 𝑖 = 𝑗)))
11080, 108, 109sylanbrc 564 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑊 = ∅) → 𝐾:ℕ–1-1𝑇)
111 f1domg 8128 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑇 ∈ Fin → (𝐾:ℕ–1-1𝑇 → ℕ ≼ 𝑇))
11279, 110, 111sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑊 = ∅) → ℕ ≼ 𝑇)
113 domfi 8336 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑇 ∈ Fin ∧ ℕ ≼ 𝑇) → ℕ ∈ Fin)
11479, 112, 113syl2anc 565 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑊 = ∅) → ℕ ∈ Fin)
115114ex 397 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑊 = ∅ → ℕ ∈ Fin))
116115necon3bd 2956 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (¬ ℕ ∈ Fin → 𝑊 ≠ ∅))
11772, 116mpi 20 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑊 ≠ ∅)
118 infssuzcl 11974 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑊 ⊆ (ℤ‘1) ∧ 𝑊 ≠ ∅) → inf(𝑊, ℝ, < ) ∈ 𝑊)
11969, 117, 118sylancr 567 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → inf(𝑊, ℝ, < ) ∈ 𝑊)
12068, 119syl5eqel 2853 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀𝑊)
12167, 120sseldi 3748 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℝ)
122121ad2antrr 697 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑀 ∈ ℝ)
12367sseli 3746 . . . . . . . . . . . . . . . . . . . 20 (𝑚𝑊𝑚 ∈ ℝ)
124123adantl 467 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑚 ∈ ℝ)
125 elfzle2 12551 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑀) → 𝑛𝑀)
126125ad2antlr 698 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑛𝑀)
127 infssuzle 11973 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑊 ⊆ (ℤ‘1) ∧ 𝑚𝑊) → inf(𝑊, ℝ, < ) ≤ 𝑚)
12869, 127mpan 662 . . . . . . . . . . . . . . . . . . . . 21 (𝑚𝑊 → inf(𝑊, ℝ, < ) ≤ 𝑚)
12968, 128syl5eqbr 4819 . . . . . . . . . . . . . . . . . . . 20 (𝑚𝑊𝑀𝑚)
130129adantl 467 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑀𝑚)
13163, 122, 124, 126, 130letrd 10395 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑚𝑊) → 𝑛𝑚)
132131ralrimiva 3114 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑚𝑊 𝑛𝑚)
13360, 132ssrabdv 3828 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑀) ⊆ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
134133adantr 466 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (1...𝑀) ⊆ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
135 simprl 746 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑖 ∈ (1...𝑀))
136134, 135sseldd 3751 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
137 simprr 748 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑗 ∈ (1...𝑀))
138134, 137sseldd 3751 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})
139136, 138jca 495 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚}))
140139, 105syldan 571 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑗))))))
14159, 140syl5ibr 236 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑗)) → 𝑖 = 𝑗))
14257, 141sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) → 𝑖 = 𝑗))
143142ralrimivva 3119 . . . . . . . . 9 (𝜑 → ∀𝑖 ∈ (1...𝑀)∀𝑗 ∈ (1...𝑀)((((𝐺𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺𝐾) ↾ (1...𝑀))‘𝑗) → 𝑖 = 𝑗))
144 dff13 6654 . . . . . . . . 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)))
14521, 143, 144sylanbrc 564 . . . . . . . 8 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1→ℕ)
146 f1f1orn 6289 . . . . . . . 8 (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1→ℕ → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺𝐾) ↾ (1...𝑀)))
147145, 146syl 17 . . . . . . 7 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺𝐾) ↾ (1...𝑀)))
148 f1oeq3 6270 . . . . . . . 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))))
1493, 148ax-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)))
150147, 149sylibr 224 . . . . . 6 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺𝐾) “ (1...𝑀)))
151 f1ofo 6285 . . . . . 6 (((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺𝐾) “ (1...𝑀)) → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺𝐾) “ (1...𝑀)))
152150, 151syl 17 . . . . 5 (𝜑 → ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺𝐾) “ (1...𝑀)))
153 fofi 8407 . . . . 5 (((1...𝑀) ∈ Fin ∧ ((𝐺𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺𝐾) “ (1...𝑀))) → ((𝐺𝐾) “ (1...𝑀)) ∈ Fin)
15443, 152, 153syl2anc 565 . . . 4 (𝜑 → ((𝐺𝐾) “ (1...𝑀)) ∈ Fin)
155 fimaxre2 11170 . . . 4 ((((𝐺𝐾) “ (1...𝑀)) ⊆ ℝ ∧ ((𝐺𝐾) “ (1...𝑀)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥)
15626, 154, 155syl2anc 565 . . 3 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦𝑥)
15742, 156r19.29a 3225 . 2 (𝜑 → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)
15897, 96resubcld 10659 . . . . 5 (𝜑 → (𝐵𝐴) ∈ ℝ)
159158rexrd 10290 . . . 4 (𝜑 → (𝐵𝐴) ∈ ℝ*)
160159adantr 466 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ∈ ℝ*)
161 fzfid 12979 . . . . . 6 (𝜑 → (1...𝑧) ∈ Fin)
162 elfznn 12576 . . . . . . . . 9 (𝑗 ∈ (1...𝑧) → 𝑗 ∈ ℕ)
163 eqid 2770 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
164163ovolfsf 23458 . . . . . . . . . . 11 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
165100, 164syl 17 . . . . . . . . . 10 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
166165ffvelrnda 6502 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞))
167162, 166sylan2 572 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞))
168 elrege0 12484 . . . . . . . 8 ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗)))
169167, 168sylib 208 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑧)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗)))
170169simpld 476 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
171161, 170fsumrecl 14672 . . . . 5 (𝜑 → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
172171adantr 466 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
173172rexrd 10290 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ*)
174163, 99ovolsf 23459 . . . . . . . . 9 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
175100, 174syl 17 . . . . . . . 8 (𝜑𝑆:ℕ⟶(0[,)+∞))
176 frn 6193 . . . . . . . 8 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
177175, 176syl 17 . . . . . . 7 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
178 rge0ssre 12486 . . . . . . 7 (0[,)+∞) ⊆ ℝ
179177, 178syl6ss 3762 . . . . . 6 (𝜑 → ran 𝑆 ⊆ ℝ)
180 ressxr 10284 . . . . . 6 ℝ ⊆ ℝ*
181179, 180syl6ss 3762 . . . . 5 (𝜑 → ran 𝑆 ⊆ ℝ*)
182 supxrcl 12349 . . . . 5 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
183181, 182syl 17 . . . 4 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
184183adantr 466 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
185158adantr 466 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ∈ ℝ)
18624sselda 3750 . . . . . . 7 ((𝜑𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑗 ∈ ℕ)
187178, 166sseldi 3748 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
188186, 187syldan 571 . . . . . 6 ((𝜑𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
189154, 188fsumrecl 14672 . . . . 5 (𝜑 → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
190189adantr 466 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
191 inss2 3980 . . . . . . . . . . 11 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
192 fss 6196 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
193100, 191, 192sylancl 566 . . . . . . . . . 10 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
19466, 120sseldi 3748 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℕ)
19515, 194ffvelrnd 6503 . . . . . . . . . . 11 (𝜑 → (𝐾𝑀) ∈ 𝑈)
1964, 195ffvelrnd 6503 . . . . . . . . . 10 (𝜑 → (𝐺‘(𝐾𝑀)) ∈ ℕ)
197193, 196ffvelrnd 6503 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐺‘(𝐾𝑀))) ∈ (ℝ × ℝ))
198 xp2nd 7347 . . . . . . . . 9 ((𝐹‘(𝐺‘(𝐾𝑀))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) ∈ ℝ)
199197, 198syl 17 . . . . . . . 8 (𝜑 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) ∈ ℝ)
20013, 8sseldi 3748 . . . . . . . . . . 11 (𝜑𝐶𝑈)
2014, 200ffvelrnd 6503 . . . . . . . . . 10 (𝜑 → (𝐺𝐶) ∈ ℕ)
202193, 201ffvelrnd 6503 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐺𝐶)) ∈ (ℝ × ℝ))
203 xp1st 7346 . . . . . . . . 9 ((𝐹‘(𝐺𝐶)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝐶))) ∈ ℝ)
204202, 203syl 17 . . . . . . . 8 (𝜑 → (1st ‘(𝐹‘(𝐺𝐶))) ∈ ℝ)
205199, 204resubcld 10659 . . . . . . 7 (𝜑 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))) ∈ ℝ)
206 fveq2 6332 . . . . . . . . . 10 (𝑗 = (𝐺‘(𝐾𝑖)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) = (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))))
207187recnd 10269 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
208186, 207syldan 571 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
209206, 43, 150, 48, 208fsumf1o 14661 . . . . . . . . 9 (𝜑 → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) = Σ𝑖 ∈ (1...𝑀)(((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))))
210100adantr 466 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2114adantr 466 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐺:𝑈⟶ℕ)
212 ffvelrn 6500 . . . . . . . . . . . . 13 ((𝐾:ℕ⟶𝑈𝑖 ∈ ℕ) → (𝐾𝑖) ∈ 𝑈)
21315, 18, 212syl2an 575 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝑈)
214211, 213ffvelrnd 6503 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘(𝐾𝑖)) ∈ ℕ)
215163ovolfsval 23457 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺‘(𝐾𝑖)) ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
216210, 214, 215syl2anc 565 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
217216sumeq2dv 14640 . . . . . . . . 9 (𝜑 → Σ𝑖 ∈ (1...𝑀)(((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾𝑖))) = Σ𝑖 ∈ (1...𝑀)((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
218193adantr 466 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
2194adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ ℕ) → 𝐺:𝑈⟶ℕ)
22015ffvelrnda 6502 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ ℕ) → (𝐾𝑖) ∈ 𝑈)
221219, 220ffvelrnd 6503 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ℕ) → (𝐺‘(𝐾𝑖)) ∈ ℕ)
222218, 221ffvelrnd 6503 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾𝑖))) ∈ (ℝ × ℝ))
223 xp2nd 7347 . . . . . . . . . . . . . 14 ((𝐹‘(𝐺‘(𝐾𝑖))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
224222, 223syl 17 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ℕ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
22518, 224sylan2 572 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
226225recnd 10269 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℂ)
227193adantr 466 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐹:ℕ⟶(ℝ × ℝ))
228227, 214ffvelrnd 6503 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐹‘(𝐺‘(𝐾𝑖))) ∈ (ℝ × ℝ))
229 xp1st 7346 . . . . . . . . . . . . 13 ((𝐹‘(𝐺‘(𝐾𝑖))) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
230228, 229syl 17 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
231230recnd 10269 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℂ)
23243, 226, 231fsumsub 14726 . . . . . . . . . 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))))))
23369, 120sseldi 3748 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (ℤ‘1))
234 fveq2 6332 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑀 → (𝐾𝑖) = (𝐾𝑀))
235234fveq2d 6336 . . . . . . . . . . . . . . 15 (𝑖 = 𝑀 → (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾𝑀)))
236235fveq2d 6336 . . . . . . . . . . . . . 14 (𝑖 = 𝑀 → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾𝑀))))
237236fveq2d 6336 . . . . . . . . . . . . 13 (𝑖 = 𝑀 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))))
238233, 226, 237fsumm1 14687 . . . . . . . . . . . 12 (𝜑 → Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) + (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀))))))
239 fzfid 12979 . . . . . . . . . . . . . . 15 (𝜑 → (1...(𝑀 − 1)) ∈ Fin)
240 elfznn 12576 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℕ)
241240, 224sylan2 572 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
242239, 241fsumrecl 14672 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℝ)
243242recnd 10269 . . . . . . . . . . . . 13 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℂ)
244199recnd 10269 . . . . . . . . . . . . 13 (𝜑 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) ∈ ℂ)
245243, 244addcomd 10439 . . . . . . . . . . . 12 (𝜑 → (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) + (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀))))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
246238, 245eqtrd 2804 . . . . . . . . . . 11 (𝜑 → Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
247 fveq2 6332 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → (𝐾𝑖) = (𝐾‘1))
248247fveq2d 6336 . . . . . . . . . . . . . . 15 (𝑖 = 1 → (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾‘1)))
249248fveq2d 6336 . . . . . . . . . . . . . 14 (𝑖 = 1 → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾‘1))))
250249fveq2d 6336 . . . . . . . . . . . . 13 (𝑖 = 1 → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘1)))))
251233, 231, 250fsum1p 14689 . . . . . . . . . . . 12 (𝜑 → Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = ((1st ‘(𝐹‘(𝐺‘(𝐾‘1)))) + Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖))))))
2525, 6, 7, 8algr0 15492 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐾‘1) = 𝐶)
253252fveq2d 6336 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺‘(𝐾‘1)) = (𝐺𝐶))
254253fveq2d 6336 . . . . . . . . . . . . . 14 (𝜑 → (𝐹‘(𝐺‘(𝐾‘1))) = (𝐹‘(𝐺𝐶)))
255254fveq2d 6336 . . . . . . . . . . . . 13 (𝜑 → (1st ‘(𝐹‘(𝐺‘(𝐾‘1)))) = (1st ‘(𝐹‘(𝐺𝐶))))
2567peano2zd 11686 . . . . . . . . . . . . . . 15 (𝜑 → (1 + 1) ∈ ℤ)
257194nnzd 11682 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℤ)
258 fzp1ss 12598 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℤ → ((1 + 1)...𝑀) ⊆ (1...𝑀))
25970, 258mp1i 13 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1 + 1)...𝑀) ⊆ (1...𝑀))
260259sselda 3750 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ ((1 + 1)...𝑀)) → 𝑖 ∈ (1...𝑀))
261260, 231syldan 571 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ((1 + 1)...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ ℂ)
262 fveq2 6332 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑗 + 1) → (𝐾𝑖) = (𝐾‘(𝑗 + 1)))
263262fveq2d 6336 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑗 + 1) → (𝐺‘(𝐾𝑖)) = (𝐺‘(𝐾‘(𝑗 + 1))))
264263fveq2d 6336 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑗 + 1) → (𝐹‘(𝐺‘(𝐾𝑖))) = (𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))))
265264fveq2d 6336 . . . . . . . . . . . . . . 15 (𝑖 = (𝑗 + 1) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))))
2667, 256, 257, 261, 265fsumshftm 14719 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))))
267 ax-1cn 10195 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
268267, 267pncan3oi 10498 . . . . . . . . . . . . . . . . 17 ((1 + 1) − 1) = 1
269268oveq1i 6802 . . . . . . . . . . . . . . . 16 (((1 + 1) − 1)...(𝑀 − 1)) = (1...(𝑀 − 1))
270269sumeq1i 14635 . . . . . . . . . . . . . . 15 Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑗 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))))
271 fvoveq1 6815 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → (𝐾‘(𝑗 + 1)) = (𝐾‘(𝑖 + 1)))
272271fveq2d 6336 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → (𝐺‘(𝐾‘(𝑗 + 1))) = (𝐺‘(𝐾‘(𝑖 + 1))))
273272fveq2d 6336 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → (𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))) = (𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
274273fveq2d 6336 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))
275274cbvsumv 14633 . . . . . . . . . . . . . . 15 Σ𝑗 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
276270, 275eqtri 2792 . . . . . . . . . . . . . 14 Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
277266, 276syl6eq 2820 . . . . . . . . . . . . 13 (𝜑 → Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))
278255, 277oveq12d 6810 . . . . . . . . . . . 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)))))))
279251, 278eqtrd 2804 . . . . . . . . . . 11 (𝜑 → Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾𝑖)))) = ((1st ‘(𝐹‘(𝐺𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
280246, 279oveq12d 6810 . . . . . . . . . 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))))))))
281204recnd 10269 . . . . . . . . . . 11 (𝜑 → (1st ‘(𝐹‘(𝐺𝐶))) ∈ ℂ)
282 peano2nn 11233 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ ℕ → (𝑖 + 1) ∈ ℕ)
283 ffvelrn 6500 . . . . . . . . . . . . . . . . . 18 ((𝐾:ℕ⟶𝑈 ∧ (𝑖 + 1) ∈ ℕ) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
28415, 282, 283syl2an 575 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ ℕ) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
285219, 284ffvelrnd 6503 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ ℕ) → (𝐺‘(𝐾‘(𝑖 + 1))) ∈ ℕ)
286218, 285ffvelrnd 6503 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))) ∈ (ℝ × ℝ))
287 xp1st 7346 . . . . . . . . . . . . . . 15 ((𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
288286, 287syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ ℕ) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
289240, 288sylan2 572 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
290239, 289fsumrecl 14672 . . . . . . . . . . . 12 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
291290recnd 10269 . . . . . . . . . . 11 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℂ)
292244, 243, 281, 291addsub4d 10640 . . . . . . . . . 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))))))))
293232, 280, 2923eqtrd 2808 . . . . . . . . 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))))))))
294209, 217, 2933eqtrd 2808 . . . . . . . 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))))))))
295294, 189eqeltrrd 2850 . . . . . . 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)
296 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑛 = 𝑀 → (𝐾𝑛) = (𝐾𝑀))
297296eleq2d 2835 . . . . . . . . . . . . . 14 (𝑛 = 𝑀 → (𝐵 ∈ (𝐾𝑛) ↔ 𝐵 ∈ (𝐾𝑀)))
298297, 64elrab2 3516 . . . . . . . . . . . . 13 (𝑀𝑊 ↔ (𝑀 ∈ ℕ ∧ 𝐵 ∈ (𝐾𝑀)))
299120, 298sylib 208 . . . . . . . . . . . 12 (𝜑 → (𝑀 ∈ ℕ ∧ 𝐵 ∈ (𝐾𝑀)))
300299simprd 477 . . . . . . . . . . 11 (𝜑𝐵 ∈ (𝐾𝑀))
30196, 97, 98, 99, 100, 73, 101, 4, 102ovolicc2lem1 23504 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐾𝑀) ∈ 𝑈) → (𝐵 ∈ (𝐾𝑀) ↔ (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑀)))) < 𝐵𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))))))
302195, 301mpdan 659 . . . . . . . . . . 11 (𝜑 → (𝐵 ∈ (𝐾𝑀) ↔ (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑀)))) < 𝐵𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))))))
303300, 302mpbid 222 . . . . . . . . . 10 (𝜑 → (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑀)))) < 𝐵𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀))))))
304303simp3d 1137 . . . . . . . . 9 (𝜑𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))))
30596, 97, 98, 99, 100, 73, 101, 4, 102ovolicc2lem1 23504 . . . . . . . . . . . 12 ((𝜑𝐶𝑈) → (𝐴𝐶 ↔ (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝐶))) < 𝐴𝐴 < (2nd ‘(𝐹‘(𝐺𝐶))))))
306200, 305mpdan 659 . . . . . . . . . . 11 (𝜑 → (𝐴𝐶 ↔ (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝐶))) < 𝐴𝐴 < (2nd ‘(𝐹‘(𝐺𝐶))))))
307104, 306mpbid 222 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝐶))) < 𝐴𝐴 < (2nd ‘(𝐹‘(𝐺𝐶)))))
308307simp2d 1136 . . . . . . . . 9 (𝜑 → (1st ‘(𝐹‘(𝐺𝐶))) < 𝐴)
30997, 204, 199, 96, 304, 308lt2subd 10852 . . . . . . . 8 (𝜑 → (𝐵𝐴) < ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))))
310158, 205, 309ltled 10386 . . . . . . 7 (𝜑 → (𝐵𝐴) ≤ ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑀)))) − (1st ‘(𝐹‘(𝐺𝐶)))))
311240adantl 467 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℕ)
312 simpr 471 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (1...(𝑀 − 1)))
313257adantr 466 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℤ)
314 elfzm11 12617 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 ∈ (1...(𝑀 − 1)) ↔ (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖𝑖 < 𝑀)))
31570, 313, 314sylancr 567 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (1...(𝑀 − 1)) ↔ (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖𝑖 < 𝑀)))
316312, 315mpbid 222 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖𝑖 < 𝑀))
317316simp3d 1137 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 < 𝑀)
318311nnred 11236 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℝ)
319121adantr 466 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℝ)
320318, 319ltnled 10385 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 < 𝑀 ↔ ¬ 𝑀𝑖))
321317, 320mpbid 222 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → ¬ 𝑀𝑖)
322 infssuzle 11973 . . . . . . . . . . . . . . . . . . . . 21 ((𝑊 ⊆ (ℤ‘1) ∧ 𝑖𝑊) → inf(𝑊, ℝ, < ) ≤ 𝑖)
32369, 322mpan 662 . . . . . . . . . . . . . . . . . . . 20 (𝑖𝑊 → inf(𝑊, ℝ, < ) ≤ 𝑖)
32468, 323syl5eqbr 4819 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑊𝑀𝑖)
325321, 324nsyl 137 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → ¬ 𝑖𝑊)
326311, 325jca 495 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ ℕ ∧ ¬ 𝑖𝑊))
32796, 97, 98, 99, 100, 73, 101, 4, 102, 11, 9, 103, 104, 8, 6, 64ovolicc2lem2 23505 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ¬ 𝑖𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵)
328326, 327syldan 571 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵)
329328iftrued 4231 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))), 𝐵) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
330 fveq2 6332 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = (𝐾𝑖) → (𝐺𝑡) = (𝐺‘(𝐾𝑖)))
331330fveq2d 6336 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝐾𝑖) → (𝐹‘(𝐺𝑡)) = (𝐹‘(𝐺‘(𝐾𝑖))))
332331fveq2d 6336 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (𝐾𝑖) → (2nd ‘(𝐹‘(𝐺𝑡))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
333332breq1d 4794 . . . . . . . . . . . . . . . . . 18 (𝑡 = (𝐾𝑖) → ((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵))
334333, 332ifbieq1d 4246 . . . . . . . . . . . . . . . . 17 (𝑡 = (𝐾𝑖) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))), 𝐵))
335 fveq2 6332 . . . . . . . . . . . . . . . . 17 (𝑡 = (𝐾𝑖) → (𝐻𝑡) = (𝐻‘(𝐾𝑖)))
336334, 335eleq12d 2843 . . . . . . . . . . . . . . . 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))))
337103ralrimiva 3114 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
338337adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
339 ffvelrn 6500 . . . . . . . . . . . . . . . . 17 ((𝐾:ℕ⟶𝑇𝑖 ∈ ℕ) → (𝐾𝑖) ∈ 𝑇)
34010, 240, 339syl2an 575 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝐾𝑖) ∈ 𝑇)
341336, 338, 340rspcdva 3464 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))), 𝐵) ∈ (𝐻‘(𝐾𝑖)))
342329, 341eqeltrrd 2850 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ (𝐻‘(𝐾𝑖)))
3435, 6, 7, 8, 9algrp1 15494 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ ℕ) → (𝐾‘(𝑖 + 1)) = (𝐻‘(𝐾𝑖)))
344240, 343sylan2 572 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝐾‘(𝑖 + 1)) = (𝐻‘(𝐾𝑖)))
345342, 344eleqtrrd 2852 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) ∈ (𝐾‘(𝑖 + 1)))
346240, 284sylan2 572 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
34796, 97, 98, 99, 100, 73, 101, 4, 102ovolicc2lem1 23504 . . . . . . . . . . . . . 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))))))))
348346, 347syldan 571 . . . . . . . . . . . . 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))))))))
349345, 348mpbid 222 . . . . . . . . . . . 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)))))))
350349simp2d 1136 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
351289, 241, 350ltled 10386 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ≤ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
352239, 289, 241, 351fsumle 14737 . . . . . . . . 9 (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ≤ Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))))
353242, 290subge0d 10818 . . . . . . . . 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))))))
354352, 353mpbird 247 . . . . . . . 8 (𝜑 → 0 ≤ (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
355242, 290resubcld 10659 . . . . . . . . 9 (𝜑 → (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))) ∈ ℝ)
356205, 355addge01d 10816 . . . . . . . 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)))))))))
357354, 356mpbid 222 . . . . . . 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))))))))
358158, 205, 295, 310, 357letrd 10395 . . . . . 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))))))))
359358, 294breqtrrd 4812 . . . . 5 (𝜑 → (𝐵𝐴) ≤ Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗))
360359adantr 466 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ≤ Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗))
361 fzfid 12979 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (1...𝑧) ∈ Fin)
362170adantlr 686 . . . . 5 (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
363169simprd 477 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑧)) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗))
364363adantlr 686 . . . . 5 (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗))
36524adantr 466 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℕ) → ((𝐺𝐾) “ (1...𝑀)) ⊆ ℕ)
366365sselda 3750 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ ℕ)
367366nnred 11236 . . . . . . . . . 10 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ ℝ)
36831ad2antlr 698 . . . . . . . . . 10 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑧 ∈ ℝ)
369 ltle 10327 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 < 𝑧𝑦𝑧))
370367, 368, 369syl2anc 565 . . . . . . . . 9 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → (𝑦 < 𝑧𝑦𝑧))
371366, 5syl6eleq 2859 . . . . . . . . . 10 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑦 ∈ (ℤ‘1))
372 nnz 11600 . . . . . . . . . . 11 (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
373372ad2antlr 698 . . . . . . . . . 10 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → 𝑧 ∈ ℤ)
374 elfz5 12540 . . . . . . . . . 10 ((𝑦 ∈ (ℤ‘1) ∧ 𝑧 ∈ ℤ) → (𝑦 ∈ (1...𝑧) ↔ 𝑦𝑧))
375371, 373, 374syl2anc 565 . . . . . . . . 9 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → (𝑦 ∈ (1...𝑧) ↔ 𝑦𝑧))
376370, 375sylibrd 249 . . . . . . . 8 (((𝜑𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))) → (𝑦 < 𝑧𝑦 ∈ (1...𝑧)))
377376ralimdva 3110 . . . . . . 7 ((𝜑𝑧 ∈ ℕ) → (∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧 → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧)))
378377impr 442 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧))
379 dfss3 3739 . . . . . 6 (((𝐺𝐾) “ (1...𝑀)) ⊆ (1...𝑧) ↔ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧))
380378, 379sylibr 224 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → ((𝐺𝐾) “ (1...𝑀)) ⊆ (1...𝑧))
381361, 362, 364, 380fsumless 14734 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ ((𝐺𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ≤ Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗))
382185, 190, 172, 360, 381letrd 10395 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ≤ Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗))
383 eqidd 2771 . . . . . 6 (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) = (((abs ∘ − ) ∘ 𝐹)‘𝑗))
384 simprl 746 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → 𝑧 ∈ ℕ)
385384, 5syl6eleq 2859 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → 𝑧 ∈ (ℤ‘1))
386362recnd 10269 . . . . . 6 (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
387383, 385, 386fsumser 14668 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑧))
38899fveq1i 6333 . . . . 5 (𝑆𝑧) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑧)
389387, 388syl6eqr 2822 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) = (𝑆𝑧))
390181adantr 466 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → ran 𝑆 ⊆ ℝ*)
391 ffn 6185 . . . . . . . 8 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
392175, 391syl 17 . . . . . . 7 (𝜑𝑆 Fn ℕ)
393392adantr 466 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → 𝑆 Fn ℕ)
394 fnfvelrn 6499 . . . . . 6 ((𝑆 Fn ℕ ∧ 𝑧 ∈ ℕ) → (𝑆𝑧) ∈ ran 𝑆)
395393, 384, 394syl2anc 565 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝑆𝑧) ∈ ran 𝑆)
396 supxrub 12358 . . . . 5 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑧) ∈ ran 𝑆) → (𝑆𝑧) ≤ sup(ran 𝑆, ℝ*, < ))
397390, 395, 396syl2anc 565 . . . 4 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝑆𝑧) ≤ sup(ran 𝑆, ℝ*, < ))
398389, 397eqbrtrd 4806 . . 3 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
399160, 173, 184, 382, 398xrletrd 12197 . 2 ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
400157, 399rexlimddv 3182 1 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  {crab 3064  cin 3720  wss 3721  c0 4061  ifcif 4223  𝒫 cpw 4295  {csn 4314   cuni 4572   class class class wbr 4784   × cxp 5247  ran crn 5250  cres 5251  cima 5252  ccom 5253   Fn wfn 6026  wf 6027  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  cdom 8106  Fincfn 8108  supcsup 8501  infcinf 8502  cc 10135  cr 10136  0cc0 10137  1c1 10138   + caddc 10140  +∞cpnf 10272  *cxr 10274   < clt 10275  cle 10276  cmin 10467  cn 11221  cz 11578  cuz 11887  (,)cioo 12379  [,)cico 12381  [,]cicc 12382  ...cfz 12532  seqcseq 13007  abscabs 14181  Σcsu 14623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-rp 12035  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-sum 14624
This theorem is referenced by:  ovolicc2lem5  23508
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