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Theorem ovolicc2lem3 23577
Description: Lemma for ovolicc2 23580. (Contributed by Mario Carneiro, 14-Jun-2014.)
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)}
Assertion
Ref Expression
ovolicc2lem3 ((𝜑 ∧ (𝑁 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝑁 = 𝑃 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑃))))))
Distinct variable groups:   𝑚,𝑛,𝑡,𝑢,𝐴   𝐵,𝑚,𝑛,𝑡,𝑢   𝑡,𝐻   𝐶,𝑚,𝑛,𝑡   𝑛,𝐹,𝑡   𝑛,𝐾,𝑡,𝑢   𝑛,𝐺,𝑡   𝑚,𝑊,𝑛   𝜑,𝑚,𝑛,𝑡   𝑇,𝑛,𝑡   𝑛,𝑁,𝑡,𝑢   𝑈,𝑛,𝑡,𝑢
Allowed substitution hints:   𝜑(𝑢)   𝐶(𝑢)   𝑃(𝑢,𝑡,𝑚,𝑛)   𝑆(𝑢,𝑡,𝑚,𝑛)   𝑇(𝑢,𝑚)   𝑈(𝑚)   𝐹(𝑢,𝑚)   𝐺(𝑢,𝑚)   𝐻(𝑢,𝑚,𝑛)   𝐾(𝑚)   𝑁(𝑚)   𝑊(𝑢,𝑡)

Proof of Theorem ovolicc2lem3
Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2fveq3 6380 . . . 4 (𝑦 = 𝑘 → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾𝑘)))
21fveq2d 6379 . . 3 (𝑦 = 𝑘 → (𝐹‘(𝐺‘(𝐾𝑦))) = (𝐹‘(𝐺‘(𝐾𝑘))))
32fveq2d 6379 . 2 (𝑦 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))
4 2fveq3 6380 . . . 4 (𝑦 = 𝑁 → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾𝑁)))
54fveq2d 6379 . . 3 (𝑦 = 𝑁 → (𝐹‘(𝐺‘(𝐾𝑦))) = (𝐹‘(𝐺‘(𝐾𝑁))))
65fveq2d 6379 . 2 (𝑦 = 𝑁 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))
7 2fveq3 6380 . . . 4 (𝑦 = 𝑃 → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾𝑃)))
87fveq2d 6379 . . 3 (𝑦 = 𝑃 → (𝐹‘(𝐺‘(𝐾𝑦))) = (𝐹‘(𝐺‘(𝐾𝑃))))
98fveq2d 6379 . 2 (𝑦 = 𝑃 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑃)))))
10 ssrab2 3847 . . 3 {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ⊆ ℕ
11 nnssre 11278 . . 3 ℕ ⊆ ℝ
1210, 11sstri 3770 . 2 {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ⊆ ℝ
1310sseli 3757 . . 3 (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} → 𝑦 ∈ ℕ)
14 ovolicc2.5 . . . . . . 7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
15 inss2 3993 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
16 fss 6236 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
1714, 15, 16sylancl 580 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
1817adantr 472 . . . . 5 ((𝜑𝑦 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
19 ovolicc2.8 . . . . . . 7 (𝜑𝐺:𝑈⟶ℕ)
2019adantr 472 . . . . . 6 ((𝜑𝑦 ∈ ℕ) → 𝐺:𝑈⟶ℕ)
21 nnuz 11923 . . . . . . . . . 10 ℕ = (ℤ‘1)
22 ovolicc2.15 . . . . . . . . . 10 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
23 1zzd 11655 . . . . . . . . . 10 (𝜑 → 1 ∈ ℤ)
24 ovolicc2.14 . . . . . . . . . 10 (𝜑𝐶𝑇)
25 ovolicc2.11 . . . . . . . . . 10 (𝜑𝐻:𝑇𝑇)
2621, 22, 23, 24, 25algrf 15569 . . . . . . . . 9 (𝜑𝐾:ℕ⟶𝑇)
2726adantr 472 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ) → 𝐾:ℕ⟶𝑇)
28 ovolicc2.10 . . . . . . . . 9 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
29 ssrab2 3847 . . . . . . . . 9 {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} ⊆ 𝑈
3028, 29eqsstri 3795 . . . . . . . 8 𝑇𝑈
31 fss 6236 . . . . . . . 8 ((𝐾:ℕ⟶𝑇𝑇𝑈) → 𝐾:ℕ⟶𝑈)
3227, 30, 31sylancl 580 . . . . . . 7 ((𝜑𝑦 ∈ ℕ) → 𝐾:ℕ⟶𝑈)
33 ffvelrn 6547 . . . . . . 7 ((𝐾:ℕ⟶𝑈𝑦 ∈ ℕ) → (𝐾𝑦) ∈ 𝑈)
3432, 33sylancom 582 . . . . . 6 ((𝜑𝑦 ∈ ℕ) → (𝐾𝑦) ∈ 𝑈)
3520, 34ffvelrnd 6550 . . . . 5 ((𝜑𝑦 ∈ ℕ) → (𝐺‘(𝐾𝑦)) ∈ ℕ)
3618, 35ffvelrnd 6550 . . . 4 ((𝜑𝑦 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾𝑦))) ∈ (ℝ × ℝ))
37 xp2nd 7399 . . . 4 ((𝐹‘(𝐺‘(𝐾𝑦))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ)
3836, 37syl 17 . . 3 ((𝜑𝑦 ∈ ℕ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ)
3913, 38sylan2 586 . 2 ((𝜑𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚}) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ)
4010sseli 3757 . . . 4 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} → 𝑘 ∈ ℕ)
4140ad2antll 720 . . 3 ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → 𝑘 ∈ ℕ)
4213anim2i 610 . . . 4 ((𝜑𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚}) → (𝜑𝑦 ∈ ℕ))
4342adantrr 708 . . 3 ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝜑𝑦 ∈ ℕ))
44 breq1 4812 . . . . . . 7 (𝑛 = 𝑘 → (𝑛𝑚𝑘𝑚))
4544ralbidv 3133 . . . . . 6 (𝑛 = 𝑘 → (∀𝑚𝑊 𝑛𝑚 ↔ ∀𝑚𝑊 𝑘𝑚))
4645elrab 3519 . . . . 5 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ↔ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 𝑘𝑚))
4746simprbi 490 . . . 4 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} → ∀𝑚𝑊 𝑘𝑚)
4847ad2antll 720 . . 3 ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → ∀𝑚𝑊 𝑘𝑚)
49 breq1 4812 . . . . . . 7 (𝑥 = 1 → (𝑥𝑚 ↔ 1 ≤ 𝑚))
5049ralbidv 3133 . . . . . 6 (𝑥 = 1 → (∀𝑚𝑊 𝑥𝑚 ↔ ∀𝑚𝑊 1 ≤ 𝑚))
51 breq2 4813 . . . . . . 7 (𝑥 = 1 → (𝑦 < 𝑥𝑦 < 1))
52 2fveq3 6380 . . . . . . . . . 10 (𝑥 = 1 → (𝐺‘(𝐾𝑥)) = (𝐺‘(𝐾‘1)))
5352fveq2d 6379 . . . . . . . . 9 (𝑥 = 1 → (𝐹‘(𝐺‘(𝐾𝑥))) = (𝐹‘(𝐺‘(𝐾‘1))))
5453fveq2d 6379 . . . . . . . 8 (𝑥 = 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))
5554breq2d 4821 . . . . . . 7 (𝑥 = 1 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))
5651, 55imbi12d 335 . . . . . 6 (𝑥 = 1 → ((𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥))))) ↔ (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))
5750, 56imbi12d 335 . . . . 5 (𝑥 = 1 → ((∀𝑚𝑊 𝑥𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))))) ↔ (∀𝑚𝑊 1 ≤ 𝑚 → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))))
5857imbi2d 331 . . . 4 (𝑥 = 1 → (((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 𝑥𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥))))))) ↔ ((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 1 ≤ 𝑚 → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))))
59 breq1 4812 . . . . . . 7 (𝑥 = 𝑘 → (𝑥𝑚𝑘𝑚))
6059ralbidv 3133 . . . . . 6 (𝑥 = 𝑘 → (∀𝑚𝑊 𝑥𝑚 ↔ ∀𝑚𝑊 𝑘𝑚))
61 breq2 4813 . . . . . . 7 (𝑥 = 𝑘 → (𝑦 < 𝑥𝑦 < 𝑘))
62 2fveq3 6380 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝐺‘(𝐾𝑥)) = (𝐺‘(𝐾𝑘)))
6362fveq2d 6379 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐹‘(𝐺‘(𝐾𝑥))) = (𝐹‘(𝐺‘(𝐾𝑘))))
6463fveq2d 6379 . . . . . . . 8 (𝑥 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))
6564breq2d 4821 . . . . . . 7 (𝑥 = 𝑘 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))
6661, 65imbi12d 335 . . . . . 6 (𝑥 = 𝑘 → ((𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥))))) ↔ (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))))
6760, 66imbi12d 335 . . . . 5 (𝑥 = 𝑘 → ((∀𝑚𝑊 𝑥𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))))) ↔ (∀𝑚𝑊 𝑘𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))))
6867imbi2d 331 . . . 4 (𝑥 = 𝑘 → (((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 𝑥𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥))))))) ↔ ((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 𝑘𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))))))
69 breq1 4812 . . . . . . 7 (𝑥 = (𝑘 + 1) → (𝑥𝑚 ↔ (𝑘 + 1) ≤ 𝑚))
7069ralbidv 3133 . . . . . 6 (𝑥 = (𝑘 + 1) → (∀𝑚𝑊 𝑥𝑚 ↔ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚))
71 breq2 4813 . . . . . . 7 (𝑥 = (𝑘 + 1) → (𝑦 < 𝑥𝑦 < (𝑘 + 1)))
72 2fveq3 6380 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝐺‘(𝐾𝑥)) = (𝐺‘(𝐾‘(𝑘 + 1))))
7372fveq2d 6379 . . . . . . . . 9 (𝑥 = (𝑘 + 1) → (𝐹‘(𝐺‘(𝐾𝑥))) = (𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))
7473fveq2d 6379 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))
7574breq2d 4821 . . . . . . 7 (𝑥 = (𝑘 + 1) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
7671, 75imbi12d 335 . . . . . 6 (𝑥 = (𝑘 + 1) → ((𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥))))) ↔ (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
7770, 76imbi12d 335 . . . . 5 (𝑥 = (𝑘 + 1) → ((∀𝑚𝑊 𝑥𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))))) ↔ (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))))
7877imbi2d 331 . . . 4 (𝑥 = (𝑘 + 1) → (((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 𝑥𝑚 → (𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥))))))) ↔ ((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))))
79 nnnlt1 11307 . . . . . . 7 (𝑦 ∈ ℕ → ¬ 𝑦 < 1)
8079adantl 473 . . . . . 6 ((𝜑𝑦 ∈ ℕ) → ¬ 𝑦 < 1)
8180pm2.21d 119 . . . . 5 ((𝜑𝑦 ∈ ℕ) → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))
8281a1d 25 . . . 4 ((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 1 ≤ 𝑚 → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))
83 nnre 11282 . . . . . . . . . . 11 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
8483adantr 472 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → 𝑘 ∈ ℝ)
8584lep1d 11209 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → 𝑘 ≤ (𝑘 + 1))
86 peano2re 10463 . . . . . . . . . . 11 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
8784, 86syl 17 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → (𝑘 + 1) ∈ ℝ)
88 ovolicc2.16 . . . . . . . . . . . . . 14 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}
89 ssrab2 3847 . . . . . . . . . . . . . 14 {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)} ⊆ ℕ
9088, 89eqsstri 3795 . . . . . . . . . . . . 13 𝑊 ⊆ ℕ
9190, 11sstri 3770 . . . . . . . . . . . 12 𝑊 ⊆ ℝ
9291sseli 3757 . . . . . . . . . . 11 (𝑚𝑊𝑚 ∈ ℝ)
9392adantl 473 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → 𝑚 ∈ ℝ)
94 letr 10385 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝑘 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑚) → 𝑘𝑚))
9584, 87, 93, 94syl3anc 1490 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → ((𝑘 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑚) → 𝑘𝑚))
9685, 95mpand 686 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → ((𝑘 + 1) ≤ 𝑚𝑘𝑚))
9796ralimdva 3109 . . . . . . 7 (𝑘 ∈ ℕ → (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → ∀𝑚𝑊 𝑘𝑚))
9897imim1d 82 . . . . . 6 (𝑘 ∈ ℕ → ((∀𝑚𝑊 𝑘𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))) → (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))))
9998adantl 473 . . . . 5 (((𝜑𝑦 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((∀𝑚𝑊 𝑘𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))) → (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))))
100 simplr 785 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑦 ∈ ℕ)
101 simprl 787 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑘 ∈ ℕ)
102 nnleltp1 11679 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑦𝑘𝑦 < (𝑘 + 1)))
103100, 101, 102syl2anc 579 . . . . . . . 8 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦𝑘𝑦 < (𝑘 + 1)))
104100nnred 11291 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑦 ∈ ℝ)
105101nnred 11291 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑘 ∈ ℝ)
106104, 105leloed 10434 . . . . . . . 8 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦𝑘 ↔ (𝑦 < 𝑘𝑦 = 𝑘)))
107103, 106bitr3d 272 . . . . . . 7 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 < (𝑘 + 1) ↔ (𝑦 < 𝑘𝑦 = 𝑘)))
108 simpll 783 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝜑)
109 ltp1 11115 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
110 ltnle 10371 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
11186, 110mpdan 678 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℝ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
112109, 111mpbid 223 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℝ → ¬ (𝑘 + 1) ≤ 𝑘)
113105, 112syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ¬ (𝑘 + 1) ≤ 𝑘)
114 breq2 4813 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → ((𝑘 + 1) ≤ 𝑚 ↔ (𝑘 + 1) ≤ 𝑘))
115114rspccv 3458 . . . . . . . . . . . . . . . . . . 19 (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑘𝑊 → (𝑘 + 1) ≤ 𝑘))
116115ad2antll 720 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑘𝑊 → (𝑘 + 1) ≤ 𝑘))
117113, 116mtod 189 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ¬ 𝑘𝑊)
118 ovolicc.1 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℝ)
119 ovolicc.2 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ ℝ)
120 ovolicc.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝐵)
121 ovolicc2.4 . . . . . . . . . . . . . . . . . 18 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
122 ovolicc2.6 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
123 ovolicc2.7 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
124 ovolicc2.9 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
125 ovolicc2.12 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
126 ovolicc2.13 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝐶)
127118, 119, 120, 121, 14, 122, 123, 19, 124, 28, 25, 125, 126, 24, 22, 88ovolicc2lem2 23576 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ ¬ 𝑘𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵)
128108, 101, 117, 127syl12anc 865 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵)
129128iftrued 4251 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))), 𝐵) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))
130 2fveq3 6380 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝐾𝑘) → (𝐹‘(𝐺𝑡)) = (𝐹‘(𝐺‘(𝐾𝑘))))
131130fveq2d 6379 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (𝐾𝑘) → (2nd ‘(𝐹‘(𝐺𝑡))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))
132131breq1d 4819 . . . . . . . . . . . . . . . . . 18 (𝑡 = (𝐾𝑘) → ((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵))
133132, 131ifbieq1d 4266 . . . . . . . . . . . . . . . . 17 (𝑡 = (𝐾𝑘) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))), 𝐵))
134 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑡 = (𝐾𝑘) → (𝐻𝑡) = (𝐻‘(𝐾𝑘)))
135133, 134eleq12d 2838 . . . . . . . . . . . . . . . 16 (𝑡 = (𝐾𝑘) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡) ↔ if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))), 𝐵) ∈ (𝐻‘(𝐾𝑘))))
136125ralrimiva 3113 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
137136ad2antrr 717 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
13826ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐾:ℕ⟶𝑇)
139138, 101ffvelrnd 6550 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾𝑘) ∈ 𝑇)
140135, 137, 139rspcdva 3467 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))), 𝐵) ∈ (𝐻‘(𝐾𝑘)))
141129, 140eqeltrrd 2845 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ (𝐻‘(𝐾𝑘)))
14221, 22, 23, 24, 25algrp1 15570 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (𝐾‘(𝑘 + 1)) = (𝐻‘(𝐾𝑘)))
143142ad2ant2r 753 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘(𝑘 + 1)) = (𝐻‘(𝐾𝑘)))
144141, 143eleqtrrd 2847 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ (𝐾‘(𝑘 + 1)))
145138, 30, 31sylancl 580 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐾:ℕ⟶𝑈)
146101peano2nnd 11293 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑘 + 1) ∈ ℕ)
147145, 146ffvelrnd 6550 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘(𝑘 + 1)) ∈ 𝑈)
148118, 119, 120, 121, 14, 122, 123, 19, 124ovolicc2lem1 23575 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐾‘(𝑘 + 1)) ∈ 𝑈) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ (𝐾‘(𝑘 + 1)) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
149108, 147, 148syl2anc 579 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ (𝐾‘(𝑘 + 1)) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
150144, 149mpbid 223 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
151150simp3d 1174 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))
15238adantr 472 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ)
15317ad2antrr 717 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐹:ℕ⟶(ℝ × ℝ))
15419ad2antrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐺:𝑈⟶ℕ)
155145, 101ffvelrnd 6550 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾𝑘) ∈ 𝑈)
156154, 155ffvelrnd 6550 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐺‘(𝐾𝑘)) ∈ ℕ)
157153, 156ffvelrnd 6550 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐹‘(𝐺‘(𝐾𝑘))) ∈ (ℝ × ℝ))
158 xp2nd 7399 . . . . . . . . . . . . 13 ((𝐹‘(𝐺‘(𝐾𝑘))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ ℝ)
159157, 158syl 17 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ ℝ)
160154, 147ffvelrnd 6550 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐺‘(𝐾‘(𝑘 + 1))) ∈ ℕ)
161153, 160ffvelrnd 6550 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))) ∈ (ℝ × ℝ))
162 xp2nd 7399 . . . . . . . . . . . . 13 ((𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ)
163161, 162syl 17 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ)
164 lttr 10368 . . . . . . . . . . . 12 (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ) → (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
165152, 159, 163, 164syl3anc 1490 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
166151, 165mpan2d 685 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
167166imim2d 57 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))) → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
168167com23 86 . . . . . . . 8 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 < 𝑘 → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
1693breq1d 4819 . . . . . . . . . 10 (𝑦 = 𝑘 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
170151, 169syl5ibrcom 238 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
171170a1dd 50 . . . . . . . 8 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 = 𝑘 → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
172168, 171jaod 885 . . . . . . 7 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ((𝑦 < 𝑘𝑦 = 𝑘) → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
173107, 172sylbid 231 . . . . . 6 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 < (𝑘 + 1) → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
174173com23 86 . . . . 5 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ((𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))) → (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))
17599, 174animpimp2impd 872 . . . 4 (𝑘 ∈ ℕ → (((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 𝑘𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))) → ((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑦 < (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))))))
17658, 68, 78, 68, 82, 175nnind 11294 . . 3 (𝑘 ∈ ℕ → ((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 𝑘𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))))
17741, 43, 48, 176syl3c 66 . 2 ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))
1783, 6, 9, 12, 39, 177eqord1 10810 1 ((𝜑 ∧ (𝑁 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝑁 = 𝑃 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑃))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  {crab 3059  cin 3731  wss 3732  c0 4079  ifcif 4243  𝒫 cpw 4315  {csn 4334   cuni 4594   class class class wbr 4809   × cxp 5275  ran crn 5278  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  Fincfn 8160  cr 10188  1c1 10190   + caddc 10192   < clt 10328  cle 10329  cmin 10520  cn 11274  (,)cioo 12377  [,]cicc 12380  seqcseq 13008  abscabs 14261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-ioo 12381  df-icc 12384  df-fz 12534  df-seq 13009
This theorem is referenced by:  ovolicc2lem4  23578
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