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Theorem ovolicc2lem3 24127
Description: Lemma for ovolicc2 24130. (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 6654 . . . 4 (𝑦 = 𝑘 → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾𝑘)))
21fveq2d 6653 . . 3 (𝑦 = 𝑘 → (𝐹‘(𝐺‘(𝐾𝑦))) = (𝐹‘(𝐺‘(𝐾𝑘))))
32fveq2d 6653 . 2 (𝑦 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))
4 2fveq3 6654 . . . 4 (𝑦 = 𝑁 → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾𝑁)))
54fveq2d 6653 . . 3 (𝑦 = 𝑁 → (𝐹‘(𝐺‘(𝐾𝑦))) = (𝐹‘(𝐺‘(𝐾𝑁))))
65fveq2d 6653 . 2 (𝑦 = 𝑁 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))
7 2fveq3 6654 . . . 4 (𝑦 = 𝑃 → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾𝑃)))
87fveq2d 6653 . . 3 (𝑦 = 𝑃 → (𝐹‘(𝐺‘(𝐾𝑦))) = (𝐹‘(𝐺‘(𝐾𝑃))))
98fveq2d 6653 . 2 (𝑦 = 𝑃 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑃)))))
10 ssrab2 4010 . . 3 {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ⊆ ℕ
11 nnssre 11633 . . 3 ℕ ⊆ ℝ
1210, 11sstri 3927 . 2 {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ⊆ ℝ
1310sseli 3914 . . 3 (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} → 𝑦 ∈ ℕ)
14 ovolicc2.5 . . . . . . 7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
15 inss2 4159 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
16 fss 6505 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
1714, 15, 16sylancl 589 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
1817adantr 484 . . . . 5 ((𝜑𝑦 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
19 ovolicc2.8 . . . . . . 7 (𝜑𝐺:𝑈⟶ℕ)
2019adantr 484 . . . . . 6 ((𝜑𝑦 ∈ ℕ) → 𝐺:𝑈⟶ℕ)
21 nnuz 12273 . . . . . . . . . 10 ℕ = (ℤ‘1)
22 ovolicc2.15 . . . . . . . . . 10 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
23 1zzd 12005 . . . . . . . . . 10 (𝜑 → 1 ∈ ℤ)
24 ovolicc2.14 . . . . . . . . . 10 (𝜑𝐶𝑇)
25 ovolicc2.11 . . . . . . . . . 10 (𝜑𝐻:𝑇𝑇)
2621, 22, 23, 24, 25algrf 15911 . . . . . . . . 9 (𝜑𝐾:ℕ⟶𝑇)
2726adantr 484 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ) → 𝐾:ℕ⟶𝑇)
28 ovolicc2.10 . . . . . . . . 9 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
2928ssrab3 4011 . . . . . . . 8 𝑇𝑈
30 fss 6505 . . . . . . . 8 ((𝐾:ℕ⟶𝑇𝑇𝑈) → 𝐾:ℕ⟶𝑈)
3127, 29, 30sylancl 589 . . . . . . 7 ((𝜑𝑦 ∈ ℕ) → 𝐾:ℕ⟶𝑈)
32 ffvelrn 6830 . . . . . . 7 ((𝐾:ℕ⟶𝑈𝑦 ∈ ℕ) → (𝐾𝑦) ∈ 𝑈)
3331, 32sylancom 591 . . . . . 6 ((𝜑𝑦 ∈ ℕ) → (𝐾𝑦) ∈ 𝑈)
3420, 33ffvelrnd 6833 . . . . 5 ((𝜑𝑦 ∈ ℕ) → (𝐺‘(𝐾𝑦)) ∈ ℕ)
3518, 34ffvelrnd 6833 . . . 4 ((𝜑𝑦 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾𝑦))) ∈ (ℝ × ℝ))
36 xp2nd 7708 . . . 4 ((𝐹‘(𝐺‘(𝐾𝑦))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ)
3735, 36syl 17 . . 3 ((𝜑𝑦 ∈ ℕ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ)
3813, 37sylan2 595 . 2 ((𝜑𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚}) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ)
3910sseli 3914 . . . 4 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} → 𝑘 ∈ ℕ)
4039ad2antll 728 . . 3 ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → 𝑘 ∈ ℕ)
4113anim2i 619 . . . 4 ((𝜑𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚}) → (𝜑𝑦 ∈ ℕ))
4241adantrr 716 . . 3 ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝜑𝑦 ∈ ℕ))
43 breq1 5036 . . . . . . 7 (𝑛 = 𝑘 → (𝑛𝑚𝑘𝑚))
4443ralbidv 3165 . . . . . 6 (𝑛 = 𝑘 → (∀𝑚𝑊 𝑛𝑚 ↔ ∀𝑚𝑊 𝑘𝑚))
4544elrab 3631 . . . . 5 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ↔ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 𝑘𝑚))
4645simprbi 500 . . . 4 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} → ∀𝑚𝑊 𝑘𝑚)
4746ad2antll 728 . . 3 ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → ∀𝑚𝑊 𝑘𝑚)
48 breq1 5036 . . . . . . 7 (𝑥 = 1 → (𝑥𝑚 ↔ 1 ≤ 𝑚))
4948ralbidv 3165 . . . . . 6 (𝑥 = 1 → (∀𝑚𝑊 𝑥𝑚 ↔ ∀𝑚𝑊 1 ≤ 𝑚))
50 breq2 5037 . . . . . . 7 (𝑥 = 1 → (𝑦 < 𝑥𝑦 < 1))
51 2fveq3 6654 . . . . . . . . . 10 (𝑥 = 1 → (𝐺‘(𝐾𝑥)) = (𝐺‘(𝐾‘1)))
5251fveq2d 6653 . . . . . . . . 9 (𝑥 = 1 → (𝐹‘(𝐺‘(𝐾𝑥))) = (𝐹‘(𝐺‘(𝐾‘1))))
5352fveq2d 6653 . . . . . . . 8 (𝑥 = 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))
5453breq2d 5045 . . . . . . 7 (𝑥 = 1 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))
5550, 54imbi12d 348 . . . . . 6 (𝑥 = 1 → ((𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥))))) ↔ (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))
5649, 55imbi12d 348 . . . . 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))))))))
5756imbi2d 344 . . . 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)))))))))
58 breq1 5036 . . . . . . 7 (𝑥 = 𝑘 → (𝑥𝑚𝑘𝑚))
5958ralbidv 3165 . . . . . 6 (𝑥 = 𝑘 → (∀𝑚𝑊 𝑥𝑚 ↔ ∀𝑚𝑊 𝑘𝑚))
60 breq2 5037 . . . . . . 7 (𝑥 = 𝑘 → (𝑦 < 𝑥𝑦 < 𝑘))
61 2fveq3 6654 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝐺‘(𝐾𝑥)) = (𝐺‘(𝐾𝑘)))
6261fveq2d 6653 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐹‘(𝐺‘(𝐾𝑥))) = (𝐹‘(𝐺‘(𝐾𝑘))))
6362fveq2d 6653 . . . . . . . 8 (𝑥 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))
6463breq2d 5045 . . . . . . 7 (𝑥 = 𝑘 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))
6560, 64imbi12d 348 . . . . . 6 (𝑥 = 𝑘 → ((𝑦 < 𝑥 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥))))) ↔ (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))))
6659, 65imbi12d 348 . . . . 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))))))))
6766imbi2d 344 . . . 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)))))))))
68 breq1 5036 . . . . . . 7 (𝑥 = (𝑘 + 1) → (𝑥𝑚 ↔ (𝑘 + 1) ≤ 𝑚))
6968ralbidv 3165 . . . . . 6 (𝑥 = (𝑘 + 1) → (∀𝑚𝑊 𝑥𝑚 ↔ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚))
70 breq2 5037 . . . . . . 7 (𝑥 = (𝑘 + 1) → (𝑦 < 𝑥𝑦 < (𝑘 + 1)))
71 2fveq3 6654 . . . . . . . . . 10 (𝑥 = (𝑘 + 1) → (𝐺‘(𝐾𝑥)) = (𝐺‘(𝐾‘(𝑘 + 1))))
7271fveq2d 6653 . . . . . . . . 9 (𝑥 = (𝑘 + 1) → (𝐹‘(𝐺‘(𝐾𝑥))) = (𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))
7372fveq2d 6653 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))
7473breq2d 5045 . . . . . . 7 (𝑥 = (𝑘 + 1) → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑥)))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
7570, 74imbi12d 348 . . . . . 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))))))))
7669, 75imbi12d 348 . . . . 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)))))))))
7776imbi2d 344 . . . 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))))))))))
78 nnnlt1 11661 . . . . . . 7 (𝑦 ∈ ℕ → ¬ 𝑦 < 1)
7978adantl 485 . . . . . 6 ((𝜑𝑦 ∈ ℕ) → ¬ 𝑦 < 1)
8079pm2.21d 121 . . . . 5 ((𝜑𝑦 ∈ ℕ) → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1))))))
8180a1d 25 . . . 4 ((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 1 ≤ 𝑚 → (𝑦 < 1 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘1)))))))
82 nnre 11636 . . . . . . . . . . 11 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
8382adantr 484 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → 𝑘 ∈ ℝ)
8483lep1d 11564 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → 𝑘 ≤ (𝑘 + 1))
85 peano2re 10806 . . . . . . . . . . 11 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
8683, 85syl 17 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → (𝑘 + 1) ∈ ℝ)
87 ovolicc2.16 . . . . . . . . . . . . . 14 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}
8887ssrab3 4011 . . . . . . . . . . . . 13 𝑊 ⊆ ℕ
8988, 11sstri 3927 . . . . . . . . . . . 12 𝑊 ⊆ ℝ
9089sseli 3914 . . . . . . . . . . 11 (𝑚𝑊𝑚 ∈ ℝ)
9190adantl 485 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → 𝑚 ∈ ℝ)
92 letr 10727 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝑘 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑚) → 𝑘𝑚))
9383, 86, 91, 92syl3anc 1368 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → ((𝑘 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑚) → 𝑘𝑚))
9484, 93mpand 694 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ 𝑚𝑊) → ((𝑘 + 1) ≤ 𝑚𝑘𝑚))
9594ralimdva 3147 . . . . . . 7 (𝑘 ∈ ℕ → (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → ∀𝑚𝑊 𝑘𝑚))
9695imim1d 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))))))))
9796adantl 485 . . . . 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))))))))
98 simplr 768 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑦 ∈ ℕ)
99 simprl 770 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑘 ∈ ℕ)
100 nnleltp1 12029 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑦𝑘𝑦 < (𝑘 + 1)))
10198, 99, 100syl2anc 587 . . . . . . . 8 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦𝑘𝑦 < (𝑘 + 1)))
10298nnred 11644 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑦 ∈ ℝ)
10399nnred 11644 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝑘 ∈ ℝ)
104102, 103leloed 10776 . . . . . . . 8 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦𝑘 ↔ (𝑦 < 𝑘𝑦 = 𝑘)))
105101, 104bitr3d 284 . . . . . . 7 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 < (𝑘 + 1) ↔ (𝑦 < 𝑘𝑦 = 𝑘)))
106 simpll 766 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝜑)
107 ltp1 11473 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
108 ltnle 10713 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
10985, 108mpdan 686 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℝ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
110107, 109mpbid 235 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℝ → ¬ (𝑘 + 1) ≤ 𝑘)
111103, 110syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ¬ (𝑘 + 1) ≤ 𝑘)
112 breq2 5037 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → ((𝑘 + 1) ≤ 𝑚 ↔ (𝑘 + 1) ≤ 𝑘))
113112rspccv 3571 . . . . . . . . . . . . . . . . . . 19 (∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚 → (𝑘𝑊 → (𝑘 + 1) ≤ 𝑘))
114113ad2antll 728 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑘𝑊 → (𝑘 + 1) ≤ 𝑘))
115111, 114mtod 201 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ¬ 𝑘𝑊)
116 ovolicc.1 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℝ)
117 ovolicc.2 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ ℝ)
118 ovolicc.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝐵)
119 ovolicc2.4 . . . . . . . . . . . . . . . . . 18 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
120 ovolicc2.6 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
121 ovolicc2.7 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
122 ovolicc2.9 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
123 ovolicc2.12 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
124 ovolicc2.13 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝐶)
125116, 117, 118, 119, 14, 120, 121, 19, 122, 28, 25, 123, 124, 24, 22, 87ovolicc2lem2 24126 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ ¬ 𝑘𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵)
126106, 99, 115, 125syl12anc 835 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵)
127126iftrued 4436 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))), 𝐵) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))
128 2fveq3 6654 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = (𝐾𝑘) → (𝐹‘(𝐺𝑡)) = (𝐹‘(𝐺‘(𝐾𝑘))))
129128fveq2d 6653 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (𝐾𝑘) → (2nd ‘(𝐹‘(𝐺𝑡))) = (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))))
130129breq1d 5043 . . . . . . . . . . . . . . . . . 18 (𝑡 = (𝐾𝑘) → ((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵))
131130, 129ifbieq1d 4451 . . . . . . . . . . . . . . . . 17 (𝑡 = (𝐾𝑘) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))), 𝐵))
132 fveq2 6649 . . . . . . . . . . . . . . . . 17 (𝑡 = (𝐾𝑘) → (𝐻𝑡) = (𝐻‘(𝐾𝑘)))
133131, 132eleq12d 2887 . . . . . . . . . . . . . . . 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))))
134123ralrimiva 3152 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
135134ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐻𝑡))
13626ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐾:ℕ⟶𝑇)
137136, 99ffvelrnd 6833 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾𝑘) ∈ 𝑇)
138133, 135, 137rspcdva 3576 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → if((2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))), 𝐵) ∈ (𝐻‘(𝐾𝑘)))
139127, 138eqeltrrd 2894 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ (𝐻‘(𝐾𝑘)))
14021, 22, 23, 24, 25algrp1 15912 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (𝐾‘(𝑘 + 1)) = (𝐻‘(𝐾𝑘)))
141140ad2ant2r 746 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘(𝑘 + 1)) = (𝐻‘(𝐾𝑘)))
142139, 141eleqtrrd 2896 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ (𝐾‘(𝑘 + 1)))
143136, 29, 30sylancl 589 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐾:ℕ⟶𝑈)
14499peano2nnd 11646 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑘 + 1) ∈ ℕ)
145143, 144ffvelrnd 6833 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾‘(𝑘 + 1)) ∈ 𝑈)
146116, 117, 118, 119, 14, 120, 121, 19, 122ovolicc2lem1 24125 . . . . . . . . . . . . . 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))))))))
147106, 145, 146syl2anc 587 . . . . . . . . . . . . 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))))))))
148142, 147mpbid 235 . . . . . . . . . . . 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)))))))
149148simp3d 1141 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))))
15037adantr 484 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) ∈ ℝ)
15117ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐹:ℕ⟶(ℝ × ℝ))
15219ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → 𝐺:𝑈⟶ℕ)
153143, 99ffvelrnd 6833 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐾𝑘) ∈ 𝑈)
154152, 153ffvelrnd 6833 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐺‘(𝐾𝑘)) ∈ ℕ)
155151, 154ffvelrnd 6833 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐹‘(𝐺‘(𝐾𝑘))) ∈ (ℝ × ℝ))
156 xp2nd 7708 . . . . . . . . . . . . 13 ((𝐹‘(𝐺‘(𝐾𝑘))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ ℝ)
157155, 156syl 17 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) ∈ ℝ)
158152, 145ffvelrnd 6833 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐺‘(𝐾‘(𝑘 + 1))) ∈ ℕ)
159151, 158ffvelrnd 6833 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))) ∈ (ℝ × ℝ))
160 xp2nd 7708 . . . . . . . . . . . . 13 ((𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ)
161159, 160syl 17 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ∈ ℝ)
162 lttr 10710 . . . . . . . . . . . 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)))))))
163150, 157, 161, 162syl3anc 1368 . . . . . . . . . . 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)))))))
164149, 163mpan2d 693 . . . . . . . . . 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)))))))
165164imim2d 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))))))))
166165com23 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))))))))
1673breq1d 5043 . . . . . . . . . 10 (𝑦 = 𝑘 → ((2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1))))) ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
168149, 167syl5ibrcom 250 . . . . . . . . 9 (((𝜑𝑦 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ ∀𝑚𝑊 (𝑘 + 1) ≤ 𝑚)) → (𝑦 = 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑘 + 1)))))))
169168a1dd 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))))))))
170166, 169jaod 856 . . . . . . 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))))))))
171105, 170sylbid 243 . . . . . 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))))))))
172171com23 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))))))))
17397, 172animpimp2impd 843 . . . 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))))))))))
17457, 67, 77, 67, 81, 173nnind 11647 . . 3 (𝑘 ∈ ℕ → ((𝜑𝑦 ∈ ℕ) → (∀𝑚𝑊 𝑘𝑚 → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))))
17540, 42, 47, 174syl3c 66 . 2 ((𝜑 ∧ (𝑦 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚} ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚𝑊 𝑛𝑚})) → (𝑦 < 𝑘 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑦)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑘))))))
1763, 6, 9, 12, 38, 175eqord1 11161 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 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  {crab 3113  cin 3883  wss 3884  c0 4246  ifcif 4428  𝒫 cpw 4500  {csn 4528   cuni 4803   class class class wbr 5033   × cxp 5521  ran crn 5524  ccom 5527  wf 6324  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Fincfn 8496  cr 10529  1c1 10531   + caddc 10533   < clt 10668  cle 10669  cmin 10863  cn 11629  (,)cioo 12730  [,]cicc 12733  seqcseq 13368  abscabs 14589
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-ioo 12734  df-icc 12737  df-fz 12890  df-seq 13369
This theorem is referenced by:  ovolicc2lem4  24128
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