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Theorem ovolicc2lem5 24104
Description: Lemma for ovolicc2 24105. (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)) ≠ ∅}
Assertion
Ref Expression
ovolicc2lem5 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   𝑢,𝑡,𝐴   𝑡,𝐵,𝑢   𝑡,𝐹   𝑡,𝐺   𝜑,𝑡   𝑡,𝑇   𝑡,𝑈,𝑢
Allowed substitution hints:   𝜑(𝑢)   𝑆(𝑢,𝑡)   𝑇(𝑢)   𝐹(𝑢)   𝐺(𝑢)

Proof of Theorem ovolicc2lem5
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolicc2.7 . . . 4 (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)
2 ovolicc.1 . . . . . 6 (𝜑𝐴 ∈ ℝ)
32rexrd 10669 . . . . 5 (𝜑𝐴 ∈ ℝ*)
4 ovolicc.2 . . . . . 6 (𝜑𝐵 ∈ ℝ)
54rexrd 10669 . . . . 5 (𝜑𝐵 ∈ ℝ*)
6 ovolicc.3 . . . . 5 (𝜑𝐴𝐵)
7 lbicc2 12833 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
83, 5, 6, 7syl3anc 1367 . . . 4 (𝜑𝐴 ∈ (𝐴[,]𝐵))
91, 8sseldd 3947 . . 3 (𝜑𝐴 𝑈)
10 eluni2 4818 . . 3 (𝐴 𝑈 ↔ ∃𝑧𝑈 𝐴𝑧)
119, 10sylib 220 . 2 (𝜑 → ∃𝑧𝑈 𝐴𝑧)
12 ovolicc2.6 . . . . . . 7 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
1312elin2d 4154 . . . . . 6 (𝜑𝑈 ∈ Fin)
14 ovolicc2.10 . . . . . . 7 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
1514ssrab3 4036 . . . . . 6 𝑇𝑈
16 ssfi 8716 . . . . . 6 ((𝑈 ∈ Fin ∧ 𝑇𝑈) → 𝑇 ∈ Fin)
1713, 15, 16sylancl 588 . . . . 5 (𝜑𝑇 ∈ Fin)
181adantr 483 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐴[,]𝐵) ⊆ 𝑈)
19 ovolicc2.8 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑈⟶ℕ)
20 ineq1 4159 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵)))
2120neeq1d 3065 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
2221, 14elrab2 3663 . . . . . . . . . . . . . . . 16 (𝑡𝑇 ↔ (𝑡𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
2322simplbi 500 . . . . . . . . . . . . . . 15 (𝑡𝑇𝑡𝑈)
24 ffvelrn 6825 . . . . . . . . . . . . . . 15 ((𝐺:𝑈⟶ℕ ∧ 𝑡𝑈) → (𝐺𝑡) ∈ ℕ)
2519, 23, 24syl2an 597 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → (𝐺𝑡) ∈ ℕ)
26 ovolicc2.5 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2726ffvelrnda 6827 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐺𝑡) ∈ ℕ) → (𝐹‘(𝐺𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2825, 27syldan 593 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝐹‘(𝐺𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2928elin2d 4154 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ))
30 xp2nd 7700 . . . . . . . . . . . 12 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
3129, 30syl 17 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
324adantr 483 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐵 ∈ ℝ)
3331, 32ifcld 4488 . . . . . . . . . 10 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ)
3422simprbi 499 . . . . . . . . . . . . . 14 (𝑡𝑇 → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
3534adantl 484 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
36 n0 4286 . . . . . . . . . . . . 13 ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
3735, 36sylib 220 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
382adantr 483 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ)
39 simprr 771 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
4039elin2d 4154 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵))
414adantr 483 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ)
42 elicc2 12781 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
432, 41, 42syl2an2r 683 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
4440, 43mpbid 234 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵))
4544simp1d 1138 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ)
4629adantrr 715 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ))
4746, 30syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
4844simp2d 1139 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴𝑦)
4939elin1d 4153 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦𝑡)
5025adantrr 715 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺𝑡) ∈ ℕ)
51 fvco3 6736 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
5226, 50, 51syl2an2r 683 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
53 ovolicc2.9 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
5423, 53sylan2 594 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡𝑇) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
5554adantrr 715 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
56 1st2nd2 7706 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺𝑡)) = ⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩)
5746, 56syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺𝑡)) = ⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩)
5857fveq2d 6650 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺𝑡))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩))
59 df-ov 7136 . . . . . . . . . . . . . . . . . . . . 21 ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩)
6058, 59syl6eqr 2873 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺𝑡))) = ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
6152, 55, 603eqtr3d 2863 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
6249, 61eleqtrd 2913 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
63 xp1st 7699 . . . . . . . . . . . . . . . . . . . 20 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
6446, 63syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
65 rexr 10665 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*)
66 rexr 10665 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*)
67 elioo2 12758 . . . . . . . . . . . . . . . . . . . 20 (((1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ* ∧ (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))))
6865, 66, 67syl2an 597 . . . . . . . . . . . . . . . . . . 19 (((1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))))
6964, 47, 68syl2anc 586 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))))
7062, 69mpbid 234 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡)))))
7170simp3d 1140 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))
7245, 47, 71ltled 10766 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
7338, 45, 47, 48, 72letrd 10775 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
7473expr 459 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡)))))
7574exlimdv 1934 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡)))))
7637, 75mpd 15 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
776adantr 483 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐴𝐵)
78 breq2 5046 . . . . . . . . . . . 12 ((2nd ‘(𝐹‘(𝐺𝑡))) = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) → (𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))) ↔ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵)))
79 breq2 5046 . . . . . . . . . . . 12 (𝐵 = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) → (𝐴𝐵𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵)))
8078, 79ifboth 4481 . . . . . . . . . . 11 ((𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))) ∧ 𝐴𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
8176, 77, 80syl2anc 586 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
82 min2 12562 . . . . . . . . . . 11 (((2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)
8331, 32, 82syl2anc 586 . . . . . . . . . 10 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)
84 elicc2 12781 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)))
852, 4, 84syl2anc 586 . . . . . . . . . . 11 (𝜑 → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)))
8685adantr 483 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)))
8733, 81, 83, 86mpbir3and 1338 . . . . . . . . 9 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
8818, 87sseldd 3947 . . . . . . . 8 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑈)
89 eluni2 4818 . . . . . . . 8 (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑈 ↔ ∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
9088, 89sylib 220 . . . . . . 7 ((𝜑𝑡𝑇) → ∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
91 simprl 769 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → 𝑥𝑈)
92 simprr 771 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
9387adantr 483 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
94 inelcm 4390 . . . . . . . . 9 ((if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
9592, 93, 94syl2anc 586 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
96 ineq1 4159 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵)))
9796neeq1d 3065 . . . . . . . . 9 (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
9897, 14elrab2 3663 . . . . . . . 8 (𝑥𝑇 ↔ (𝑥𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
9991, 95, 98sylanbrc 585 . . . . . . 7 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → 𝑥𝑇)
10090, 99, 92reximssdv 3263 . . . . . 6 ((𝜑𝑡𝑇) → ∃𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
101100ralrimiva 3169 . . . . 5 (𝜑 → ∀𝑡𝑇𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
102 eleq2 2899 . . . . . 6 (𝑥 = (𝑡) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥 ↔ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
103102ac6sfi 8740 . . . . 5 ((𝑇 ∈ Fin ∧ ∀𝑡𝑇𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥) → ∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
10417, 101, 103syl2anc 586 . . . 4 (𝜑 → ∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
105104adantr 483 . . 3 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → ∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
106 2fveq3 6651 . . . . . . . . . . 11 (𝑥 = 𝑡 → (𝐹‘(𝐺𝑥)) = (𝐹‘(𝐺𝑡)))
107106fveq2d 6650 . . . . . . . . . 10 (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺𝑥))) = (2nd ‘(𝐹‘(𝐺𝑡))))
108107breq1d 5052 . . . . . . . . 9 (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵))
109108, 107ifbieq1d 4466 . . . . . . . 8 (𝑥 = 𝑡 → if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
110 fveq2 6646 . . . . . . . 8 (𝑥 = 𝑡 → (𝑥) = (𝑡))
111109, 110eleq12d 2905 . . . . . . 7 (𝑥 = 𝑡 → (if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ↔ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
112111cbvralvw 3428 . . . . . 6 (∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ↔ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡))
1132adantr 483 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴 ∈ ℝ)
1144adantr 483 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐵 ∈ ℝ)
1156adantr 483 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴𝐵)
116 ovolicc2.4 . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
11726adantr 483 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
11812adantr 483 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
1191adantr 483 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝐴[,]𝐵) ⊆ 𝑈)
12019adantr 483 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐺:𝑈⟶ℕ)
12153adantlr 713 . . . . . . . . 9 (((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) ∧ 𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
122 simprrl 779 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → :𝑇𝑇)
123 simprrr 780 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥))
124111rspccva 3601 . . . . . . . . . 10 ((∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ∧ 𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡))
125123, 124sylan 582 . . . . . . . . 9 (((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) ∧ 𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡))
126 simprlr 778 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴𝑧)
127 simprll 777 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑧𝑈)
1288adantr 483 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴 ∈ (𝐴[,]𝐵))
129 inelcm 4390 . . . . . . . . . . 11 ((𝐴𝑧𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
130126, 128, 129syl2anc 586 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
131 ineq1 4159 . . . . . . . . . . . 12 (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵)))
132131neeq1d 3065 . . . . . . . . . . 11 (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
133132, 14elrab2 3663 . . . . . . . . . 10 (𝑧𝑇 ↔ (𝑧𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
134127, 130, 133sylanbrc 585 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑧𝑇)
135 eqid 2820 . . . . . . . . 9 seq1(( ∘ 1st ), (ℕ × {𝑧})) = seq1(( ∘ 1st ), (ℕ × {𝑧}))
136 fveq2 6646 . . . . . . . . . . 11 (𝑚 = 𝑛 → (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚) = (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛))
137136eleq2d 2896 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛)))
138137cbvrabv 3470 . . . . . . . . 9 {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛)}
139 eqid 2820 . . . . . . . . 9 inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < )
140113, 114, 115, 116, 117, 118, 119, 120, 121, 14, 122, 125, 126, 134, 135, 138, 139ovolicc2lem4 24103 . . . . . . . 8 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
141140anassrs 470 . . . . . . 7 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥))) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
142141expr 459 . . . . . 6 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ :𝑇𝑇) → (∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
143112, 142syl5bir 245 . . . . 5 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ :𝑇𝑇) → (∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
144143expimpd 456 . . . 4 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → ((:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
145144exlimdv 1934 . . 3 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → (∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
146105, 145mpd 15 . 2 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
14711, 146rexlimddv 3278 1 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wex 1780  wcel 2114  wne 3006  wral 3125  wrex 3126  {crab 3129  cin 3912  wss 3913  c0 4269  ifcif 4443  𝒫 cpw 4515  {csn 4543  cop 4549   cuni 4814   class class class wbr 5042   × cxp 5529  ran crn 5532  ccom 5535  wf 6327  cfv 6331  (class class class)co 7133  1st c1st 7665  2nd c2nd 7666  Fincfn 8487  supcsup 8882  infcinf 8883  cr 10514  1c1 10516   + caddc 10518  *cxr 10652   < clt 10653  cle 10654  cmin 10848  cn 11616  (,)cioo 12717  [,]cicc 12720  seqcseq 13353  abscabs 14573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-sup 8884  df-inf 8885  df-oi 8952  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-n0 11877  df-z 11961  df-uz 12223  df-rp 12369  df-ioo 12721  df-ico 12723  df-icc 12724  df-fz 12877  df-fzo 13018  df-seq 13354  df-exp 13415  df-hash 13676  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-clim 14825  df-sum 15023
This theorem is referenced by:  ovolicc2  24105
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