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Theorem ovolicc2lem5 25556
Description: Lemma for ovolicc2 25557. (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 11311 . . . . 5 (𝜑𝐴 ∈ ℝ*)
4 ovolicc.2 . . . . . 6 (𝜑𝐵 ∈ ℝ)
54rexrd 11311 . . . . 5 (𝜑𝐵 ∈ ℝ*)
6 ovolicc.3 . . . . 5 (𝜑𝐴𝐵)
7 lbicc2 13504 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
83, 5, 6, 7syl3anc 1373 . . . 4 (𝜑𝐴 ∈ (𝐴[,]𝐵))
91, 8sseldd 3984 . . 3 (𝜑𝐴 𝑈)
10 eluni2 4911 . . 3 (𝐴 𝑈 ↔ ∃𝑧𝑈 𝐴𝑧)
119, 10sylib 218 . 2 (𝜑 → ∃𝑧𝑈 𝐴𝑧)
12 ovolicc2.6 . . . . . . 7 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
1312elin2d 4205 . . . . . 6 (𝜑𝑈 ∈ Fin)
14 ovolicc2.10 . . . . . . 7 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
1514ssrab3 4082 . . . . . 6 𝑇𝑈
16 ssfi 9213 . . . . . 6 ((𝑈 ∈ Fin ∧ 𝑇𝑈) → 𝑇 ∈ Fin)
1713, 15, 16sylancl 586 . . . . 5 (𝜑𝑇 ∈ Fin)
181adantr 480 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐴[,]𝐵) ⊆ 𝑈)
19 ovolicc2.8 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑈⟶ℕ)
20 ineq1 4213 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵)))
2120neeq1d 3000 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
2221, 14elrab2 3695 . . . . . . . . . . . . . . . 16 (𝑡𝑇 ↔ (𝑡𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
2322simplbi 497 . . . . . . . . . . . . . . 15 (𝑡𝑇𝑡𝑈)
24 ffvelcdm 7101 . . . . . . . . . . . . . . 15 ((𝐺:𝑈⟶ℕ ∧ 𝑡𝑈) → (𝐺𝑡) ∈ ℕ)
2519, 23, 24syl2an 596 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → (𝐺𝑡) ∈ ℕ)
26 ovolicc2.5 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2726ffvelcdmda 7104 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐺𝑡) ∈ ℕ) → (𝐹‘(𝐺𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2825, 27syldan 591 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝐹‘(𝐺𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2928elin2d 4205 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ))
30 xp2nd 8047 . . . . . . . . . . . 12 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
3129, 30syl 17 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
324adantr 480 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐵 ∈ ℝ)
3331, 32ifcld 4572 . . . . . . . . . 10 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ ℝ)
3422simprbi 496 . . . . . . . . . . . . . 14 (𝑡𝑇 → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
3534adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)
36 n0 4353 . . . . . . . . . . . . 13 ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
3735, 36sylib 218 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
382adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ)
39 simprr 773 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
4039elin2d 4205 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵))
414adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ)
42 elicc2 13452 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
432, 41, 42syl2an2r 685 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
4440, 43mpbid 232 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵))
4544simp1d 1143 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ)
4629adantrr 717 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ))
4746, 30syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
4844simp2d 1144 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴𝑦)
4939elin1d 4204 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦𝑡)
5025adantrr 717 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺𝑡) ∈ ℕ)
51 fvco3 7008 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
5226, 50, 51syl2an2r 685 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
53 ovolicc2.9 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
5423, 53sylan2 593 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡𝑇) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
5554adantrr 717 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
56 1st2nd2 8053 . . . . . . . . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺𝑡))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩))
59 df-ov 7434 . . . . . . . . . . . . . . . . . . . . 21 ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩)
6058, 59eqtr4di 2795 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺𝑡))) = ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
6152, 55, 603eqtr3d 2785 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
6249, 61eleqtrd 2843 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
63 xp1st 8046 . . . . . . . . . . . . . . . . . . . 20 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
6446, 63syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
65 rexr 11307 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*)
66 rexr 11307 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*)
67 elioo2 13428 . . . . . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))))
7062, 69mpbid 232 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡)))))
7170simp3d 1145 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))
7245, 47, 71ltled 11409 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
7338, 45, 47, 48, 72letrd 11418 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
7473expr 456 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡)))))
7574exlimdv 1933 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡)))))
7637, 75mpd 15 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
776adantr 480 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → 𝐴𝐵)
78 breq2 5147 . . . . . . . . . . . 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 5147 . . . . . . . . . . . 12 (𝐵 = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) → (𝐴𝐵𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵)))
8078, 79ifboth 4565 . . . . . . . . . . 11 ((𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))) ∧ 𝐴𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
8176, 77, 80syl2anc 584 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
82 min2 13232 . . . . . . . . . . 11 (((2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)
8331, 32, 82syl2anc 584 . . . . . . . . . 10 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)
84 elicc2 13452 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 480 . . . . . . . . . 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 1343 . . . . . . . . 9 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
8818, 87sseldd 3984 . . . . . . . 8 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑈)
89 eluni2 4911 . . . . . . . 8 (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑈 ↔ ∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
9088, 89sylib 218 . . . . . . 7 ((𝜑𝑡𝑇) → ∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
91 simprl 771 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → 𝑥𝑈)
92 simprr 773 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
9387adantr 480 . . . . . . . . 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 4465 . . . . . . . . 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 584 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
96 ineq1 4213 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵)))
9796neeq1d 3000 . . . . . . . . 9 (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
9897, 14elrab2 3695 . . . . . . . 8 (𝑥𝑇 ↔ (𝑥𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
9991, 95, 98sylanbrc 583 . . . . . . 7 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → 𝑥𝑇)
10090, 99, 92reximssdv 3173 . . . . . 6 ((𝜑𝑡𝑇) → ∃𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
101100ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑡𝑇𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
102 eleq2 2830 . . . . . 6 (𝑥 = (𝑡) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥 ↔ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
103102ac6sfi 9320 . . . . 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 584 . . . 4 (𝜑 → ∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
105104adantr 480 . . 3 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → ∃(:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
106 2fveq3 6911 . . . . . . . . . . 11 (𝑥 = 𝑡 → (𝐹‘(𝐺𝑥)) = (𝐹‘(𝐺𝑡)))
107106fveq2d 6910 . . . . . . . . . 10 (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺𝑥))) = (2nd ‘(𝐹‘(𝐺𝑡))))
108107breq1d 5153 . . . . . . . . 9 (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵))
109108, 107ifbieq1d 4550 . . . . . . . 8 (𝑥 = 𝑡 → if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
110 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑡 → (𝑥) = (𝑡))
111109, 110eleq12d 2835 . . . . . . 7 (𝑥 = 𝑡 → (if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ↔ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
112111cbvralvw 3237 . . . . . 6 (∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ↔ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡))
1132adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴 ∈ ℝ)
1144adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐵 ∈ ℝ)
1156adantr 480 . . . . . . . . 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 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
11812adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
1191adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝐴[,]𝐵) ⊆ 𝑈)
12019adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐺:𝑈⟶ℕ)
12153adantlr 715 . . . . . . . . 9 (((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) ∧ 𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
122 simprrl 781 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → :𝑇𝑇)
123 simprrr 782 . . . . . . . . . 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 3621 . . . . . . . . . 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 580 . . . . . . . . 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 780 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴𝑧)
127 simprll 779 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑧𝑈)
1288adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴 ∈ (𝐴[,]𝐵))
129 inelcm 4465 . . . . . . . . . . 11 ((𝐴𝑧𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
130126, 128, 129syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
131 ineq1 4213 . . . . . . . . . . . 12 (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵)))
132131neeq1d 3000 . . . . . . . . . . 11 (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
133132, 14elrab2 3695 . . . . . . . . . 10 (𝑧𝑇 ↔ (𝑧𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
134127, 130, 133sylanbrc 583 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑧𝑇)
135 eqid 2737 . . . . . . . . 9 seq1(( ∘ 1st ), (ℕ × {𝑧})) = seq1(( ∘ 1st ), (ℕ × {𝑧}))
136 fveq2 6906 . . . . . . . . . . 11 (𝑚 = 𝑛 → (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚) = (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛))
137136eleq2d 2827 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛)))
138137cbvrabv 3447 . . . . . . . . 9 {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛)}
139 eqid 2737 . . . . . . . . 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 25555 . . . . . . . 8 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
141140anassrs 467 . . . . . . 7 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥))) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
142141expr 456 . . . . . 6 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ :𝑇𝑇) → (∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
143112, 142biimtrrid 243 . . . . 5 (((𝜑 ∧ (𝑧𝑈𝐴𝑧)) ∧ :𝑇𝑇) → (∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
144143expimpd 453 . . . 4 ((𝜑 ∧ (𝑧𝑈𝐴𝑧)) → ((:𝑇𝑇 ∧ ∀𝑡𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)) → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < )))
145144exlimdv 1933 . . 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 3161 1 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  cin 3950  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600  {csn 4626  cop 4632   cuni 4907   class class class wbr 5143   × cxp 5683  ran crn 5686  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Fincfn 8985  supcsup 9480  infcinf 9481  cr 11154  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cle 11296  cmin 11492  cn 12266  (,)cioo 13387  [,]cicc 13390  seqcseq 14042  abscabs 15273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723
This theorem is referenced by:  ovolicc2  25557
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