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Theorem ovolicc2lem2 23037
Description: Lemma for ovolicc2 23041. (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
ovolicc2lem2 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ ¬ 𝑁𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ≤ 𝐵)
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,𝑊   𝜑,𝑛,𝑡   𝑇,𝑛,𝑡   𝑛,𝑁,𝑡,𝑢   𝑈,𝑛,𝑡,𝑢
Allowed substitution hints:   𝜑(𝑢)   𝐶(𝑢)   𝑆(𝑢,𝑡,𝑛)   𝑇(𝑢)   𝐹(𝑢)   𝐺(𝑢)   𝐻(𝑢,𝑛)   𝑊(𝑢,𝑡)

Proof of Theorem ovolicc2lem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovolicc.2 . . . . . 6 (𝜑𝐵 ∈ ℝ)
21adantr 479 . . . . 5 ((𝜑𝑁 ∈ ℕ) → 𝐵 ∈ ℝ)
3 ovolicc2.5 . . . . . . . . 9 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4 inss2 3795 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
5 fss 5954 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
63, 4, 5sylancl 692 . . . . . . . 8 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
76adantr 479 . . . . . . 7 ((𝜑𝑁 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
8 ovolicc2.8 . . . . . . . . 9 (𝜑𝐺:𝑈⟶ℕ)
98adantr 479 . . . . . . . 8 ((𝜑𝑁 ∈ ℕ) → 𝐺:𝑈⟶ℕ)
10 nnuz 11557 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
11 ovolicc2.15 . . . . . . . . . . . 12 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
12 1zzd 11243 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℤ)
13 ovolicc2.14 . . . . . . . . . . . 12 (𝜑𝐶𝑇)
14 ovolicc2.11 . . . . . . . . . . . 12 (𝜑𝐻:𝑇𝑇)
1510, 11, 12, 13, 14algrf 15072 . . . . . . . . . . 11 (𝜑𝐾:ℕ⟶𝑇)
1615ffvelrnda 6251 . . . . . . . . . 10 ((𝜑𝑁 ∈ ℕ) → (𝐾𝑁) ∈ 𝑇)
17 ineq1 3768 . . . . . . . . . . . 12 (𝑢 = (𝐾𝑁) → (𝑢 ∩ (𝐴[,]𝐵)) = ((𝐾𝑁) ∩ (𝐴[,]𝐵)))
1817neeq1d 2840 . . . . . . . . . . 11 (𝑢 = (𝐾𝑁) → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ((𝐾𝑁) ∩ (𝐴[,]𝐵)) ≠ ∅))
19 ovolicc2.10 . . . . . . . . . . 11 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
2018, 19elrab2 3332 . . . . . . . . . 10 ((𝐾𝑁) ∈ 𝑇 ↔ ((𝐾𝑁) ∈ 𝑈 ∧ ((𝐾𝑁) ∩ (𝐴[,]𝐵)) ≠ ∅))
2116, 20sylib 206 . . . . . . . . 9 ((𝜑𝑁 ∈ ℕ) → ((𝐾𝑁) ∈ 𝑈 ∧ ((𝐾𝑁) ∩ (𝐴[,]𝐵)) ≠ ∅))
2221simpld 473 . . . . . . . 8 ((𝜑𝑁 ∈ ℕ) → (𝐾𝑁) ∈ 𝑈)
239, 22ffvelrnd 6252 . . . . . . 7 ((𝜑𝑁 ∈ ℕ) → (𝐺‘(𝐾𝑁)) ∈ ℕ)
247, 23ffvelrnd 6252 . . . . . 6 ((𝜑𝑁 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾𝑁))) ∈ (ℝ × ℝ))
25 xp2nd 7067 . . . . . 6 ((𝐹‘(𝐺‘(𝐾𝑁))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ∈ ℝ)
2624, 25syl 17 . . . . 5 ((𝜑𝑁 ∈ ℕ) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ∈ ℝ)
272, 26ltnled 10035 . . . 4 ((𝜑𝑁 ∈ ℕ) → (𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ↔ ¬ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ≤ 𝐵))
28 simprl 789 . . . . . 6 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → 𝑁 ∈ ℕ)
291adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → 𝐵 ∈ ℝ)
3021adantrr 748 . . . . . . . . . 10 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → ((𝐾𝑁) ∈ 𝑈 ∧ ((𝐾𝑁) ∩ (𝐴[,]𝐵)) ≠ ∅))
3130simprd 477 . . . . . . . . 9 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → ((𝐾𝑁) ∩ (𝐴[,]𝐵)) ≠ ∅)
32 n0 3889 . . . . . . . . 9 (((𝐾𝑁) ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵)))
3331, 32sylib 206 . . . . . . . 8 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → ∃𝑥 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵)))
34 xp1st 7066 . . . . . . . . . . . 12 ((𝐹‘(𝐺‘(𝐾𝑁))) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ∈ ℝ)
3524, 34syl 17 . . . . . . . . . . 11 ((𝜑𝑁 ∈ ℕ) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ∈ ℝ)
3635adantrr 748 . . . . . . . . . 10 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ∈ ℝ)
3736adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ∈ ℝ)
38 simpr 475 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵)))
39 elin 3757 . . . . . . . . . . . . 13 (𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵)) ↔ (𝑥 ∈ (𝐾𝑁) ∧ 𝑥 ∈ (𝐴[,]𝐵)))
4038, 39sylib 206 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → (𝑥 ∈ (𝐾𝑁) ∧ 𝑥 ∈ (𝐴[,]𝐵)))
4140simprd 477 . . . . . . . . . . 11 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐴[,]𝐵))
42 ovolicc.1 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℝ)
43 elicc2 12067 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
4442, 1, 43syl2anc 690 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
4544ad2antrr 757 . . . . . . . . . . 11 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
4641, 45mpbid 220 . . . . . . . . . 10 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
4746simp1d 1065 . . . . . . . . 9 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → 𝑥 ∈ ℝ)
481ad2antrr 757 . . . . . . . . 9 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → 𝐵 ∈ ℝ)
4940simpld 473 . . . . . . . . . . 11 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → 𝑥 ∈ (𝐾𝑁))
5030simpld 473 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → (𝐾𝑁) ∈ 𝑈)
51 ovolicc.3 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
52 ovolicc2.4 . . . . . . . . . . . . . 14 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
53 ovolicc2.6 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
54 ovolicc2.7 . . . . . . . . . . . . . 14 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
55 ovolicc2.9 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
5642, 1, 51, 52, 3, 53, 54, 8, 55ovolicc2lem1 23036 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐾𝑁) ∈ 𝑈) → (𝑥 ∈ (𝐾𝑁) ↔ (𝑥 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝑥𝑥 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))))
5750, 56syldan 485 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → (𝑥 ∈ (𝐾𝑁) ↔ (𝑥 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝑥𝑥 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))))
5857adantr 479 . . . . . . . . . . 11 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → (𝑥 ∈ (𝐾𝑁) ↔ (𝑥 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝑥𝑥 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))))
5949, 58mpbid 220 . . . . . . . . . 10 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → (𝑥 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝑥𝑥 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁))))))
6059simp2d 1066 . . . . . . . . 9 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝑥)
6146simp3d 1067 . . . . . . . . 9 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → 𝑥𝐵)
6237, 47, 48, 60, 61ltletrd 10048 . . . . . . . 8 (((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) ∧ 𝑥 ∈ ((𝐾𝑁) ∩ (𝐴[,]𝐵))) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝐵)
6333, 62exlimddv 1849 . . . . . . 7 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝐵)
64 simprr 791 . . . . . . 7 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))
6542, 1, 51, 52, 3, 53, 54, 8, 55ovolicc2lem1 23036 . . . . . . . 8 ((𝜑 ∧ (𝐾𝑁) ∈ 𝑈) → (𝐵 ∈ (𝐾𝑁) ↔ (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝐵𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))))
6650, 65syldan 485 . . . . . . 7 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → (𝐵 ∈ (𝐾𝑁) ↔ (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾𝑁)))) < 𝐵𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))))
6729, 63, 64, 66mpbir3and 1237 . . . . . 6 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → 𝐵 ∈ (𝐾𝑁))
68 fveq2 6087 . . . . . . . 8 (𝑛 = 𝑁 → (𝐾𝑛) = (𝐾𝑁))
6968eleq2d 2672 . . . . . . 7 (𝑛 = 𝑁 → (𝐵 ∈ (𝐾𝑛) ↔ 𝐵 ∈ (𝐾𝑁)))
70 ovolicc2.16 . . . . . . 7 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾𝑛)}
7169, 70elrab2 3332 . . . . . 6 (𝑁𝑊 ↔ (𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝐾𝑁)))
7228, 67, 71sylanbrc 694 . . . . 5 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))))) → 𝑁𝑊)
7372expr 640 . . . 4 ((𝜑𝑁 ∈ ℕ) → (𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) → 𝑁𝑊))
7427, 73sylbird 248 . . 3 ((𝜑𝑁 ∈ ℕ) → (¬ (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ≤ 𝐵𝑁𝑊))
7574con1d 137 . 2 ((𝜑𝑁 ∈ ℕ) → (¬ 𝑁𝑊 → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ≤ 𝐵))
7675impr 646 1 ((𝜑 ∧ (𝑁 ∈ ℕ ∧ ¬ 𝑁𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾𝑁)))) ≤ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wne 2779  {crab 2899  cin 3538  wss 3539  c0 3873  ifcif 4035  𝒫 cpw 4107  {csn 4124   cuni 4366   class class class wbr 4577   × cxp 5025  ran crn 5028  ccom 5031  wf 5785  cfv 5789  (class class class)co 6526  1st c1st 7034  2nd c2nd 7035  Fincfn 7818  cr 9791  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117  cn 10869  (,)cioo 12004  [,]cicc 12007  seqcseq 12620  abscabs 13770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-n0 11142  df-z 11213  df-uz 11522  df-ioo 12008  df-icc 12011  df-fz 12155  df-seq 12621
This theorem is referenced by:  ovolicc2lem3  23038  ovolicc2lem4  23039
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