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Theorem ovolicc2lem5 25283
Description: Lemma for ovolicc2 25284. (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 11271 . . . . 5 (𝜑𝐴 ∈ ℝ*)
4 ovolicc.2 . . . . . 6 (𝜑𝐵 ∈ ℝ)
54rexrd 11271 . . . . 5 (𝜑𝐵 ∈ ℝ*)
6 ovolicc.3 . . . . 5 (𝜑𝐴𝐵)
7 lbicc2 13448 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
83, 5, 6, 7syl3anc 1370 . . . 4 (𝜑𝐴 ∈ (𝐴[,]𝐵))
91, 8sseldd 3983 . . 3 (𝜑𝐴 𝑈)
10 eluni2 4912 . . 3 (𝐴 𝑈 ↔ ∃𝑧𝑈 𝐴𝑧)
119, 10sylib 217 . 2 (𝜑 → ∃𝑧𝑈 𝐴𝑧)
12 ovolicc2.6 . . . . . . 7 (𝜑𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
1312elin2d 4199 . . . . . 6 (𝜑𝑈 ∈ Fin)
14 ovolicc2.10 . . . . . . 7 𝑇 = {𝑢𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
1514ssrab3 4080 . . . . . 6 𝑇𝑈
16 ssfi 9179 . . . . . 6 ((𝑈 ∈ Fin ∧ 𝑇𝑈) → 𝑇 ∈ Fin)
1713, 15, 16sylancl 585 . . . . 5 (𝜑𝑇 ∈ Fin)
181adantr 480 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐴[,]𝐵) ⊆ 𝑈)
19 ovolicc2.8 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑈⟶ℕ)
20 ineq1 4205 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵)))
2120neeq1d 2999 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
2221, 14elrab2 3686 . . . . . . . . . . . . . . . 16 (𝑡𝑇 ↔ (𝑡𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅))
2322simplbi 497 . . . . . . . . . . . . . . 15 (𝑡𝑇𝑡𝑈)
24 ffvelcdm 7083 . . . . . . . . . . . . . . 15 ((𝐺:𝑈⟶ℕ ∧ 𝑡𝑈) → (𝐺𝑡) ∈ ℕ)
2519, 23, 24syl2an 595 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → (𝐺𝑡) ∈ ℕ)
26 ovolicc2.5 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2726ffvelcdmda 7086 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐺𝑡) ∈ ℕ) → (𝐹‘(𝐺𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2825, 27syldan 590 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝐹‘(𝐺𝑡)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2928elin2d 4199 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ))
30 xp2nd 8012 . . . . . . . . . . . 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 4574 . . . . . . . . . 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 4346 . . . . . . . . . . . . 13 ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
3735, 36sylib 217 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
382adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ)
39 simprr 770 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))
4039elin2d 4199 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵))
414adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ)
42 elicc2 13396 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
432, 41, 42syl2an2r 682 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵)))
4440, 43mpbid 231 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴𝑦𝑦𝐵))
4544simp1d 1141 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ)
4629adantrr 714 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ))
4746, 30syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
4844simp2d 1142 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴𝑦)
4939elin1d 4198 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦𝑡)
5025adantrr 714 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺𝑡) ∈ ℕ)
51 fvco3 6990 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
5226, 50, 51syl2an2r 682 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = ((,)‘(𝐹‘(𝐺𝑡))))
53 ovolicc2.9 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
5423, 53sylan2 592 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡𝑇) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
5554adantrr 714 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
56 1st2nd2 8018 . . . . . . . . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺𝑡))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩))
59 df-ov 7415 . . . . . . . . . . . . . . . . . . . . 21 ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺𝑡))), (2nd ‘(𝐹‘(𝐺𝑡)))⟩)
6058, 59eqtr4di 2789 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺𝑡))) = ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
6152, 55, 603eqtr3d 2779 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
6249, 61eleqtrd 2834 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))))
63 xp1st 8011 . . . . . . . . . . . . . . . . . . . 20 ((𝐹‘(𝐺𝑡)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
6446, 63syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ)
65 rexr 11267 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ → (1st ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*)
66 rexr 11267 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ*)
67 elioo2 13372 . . . . . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺𝑡)))(,)(2nd ‘(𝐹‘(𝐺𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))))
7062, 69mpbid 231 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺𝑡))) < 𝑦𝑦 < (2nd ‘(𝐹‘(𝐺𝑡)))))
7170simp3d 1143 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺𝑡))))
7245, 47, 71ltled 11369 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
7338, 45, 47, 48, 72letrd 11378 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑡𝑇𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))))
7473expr 456 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡)))))
7574exlimdv 1935 . . . . . . . . . . . 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 5152 . . . . . . . . . . . 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 5152 . . . . . . . . . . . 12 (𝐵 = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) → (𝐴𝐵𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵)))
8078, 79ifboth 4567 . . . . . . . . . . 11 ((𝐴 ≤ (2nd ‘(𝐹‘(𝐺𝑡))) ∧ 𝐴𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
8176, 77, 80syl2anc 583 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
82 min2 13176 . . . . . . . . . . 11 (((2nd ‘(𝐹‘(𝐺𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)
8331, 32, 82syl2anc 583 . . . . . . . . . 10 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ≤ 𝐵)
84 elicc2 13396 . . . . . . . . . . . 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 583 . . . . . . . . . . 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 1341 . . . . . . . . 9 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝐴[,]𝐵))
8818, 87sseldd 3983 . . . . . . . 8 ((𝜑𝑡𝑇) → if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑈)
89 eluni2 4912 . . . . . . . 8 (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑈 ↔ ∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
9088, 89sylib 217 . . . . . . 7 ((𝜑𝑡𝑇) → ∃𝑥𝑈 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
91 simprl 768 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → 𝑥𝑈)
92 simprr 770 . . . . . . . . 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 4464 . . . . . . . . 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 583 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)
96 ineq1 4205 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵)))
9796neeq1d 2999 . . . . . . . . 9 (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
9897, 14elrab2 3686 . . . . . . . 8 (𝑥𝑇 ↔ (𝑥𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅))
9991, 95, 98sylanbrc 582 . . . . . . 7 (((𝜑𝑡𝑇) ∧ (𝑥𝑈 ∧ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)) → 𝑥𝑇)
10090, 99, 92reximssdv 3171 . . . . . 6 ((𝜑𝑡𝑇) → ∃𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
101100ralrimiva 3145 . . . . 5 (𝜑 → ∀𝑡𝑇𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥)
102 eleq2 2821 . . . . . 6 (𝑥 = (𝑡) → (if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ 𝑥 ↔ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
103102ac6sfi 9293 . . . . 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 583 . . . 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 6896 . . . . . . . . . . 11 (𝑥 = 𝑡 → (𝐹‘(𝐺𝑥)) = (𝐹‘(𝐺𝑡)))
107106fveq2d 6895 . . . . . . . . . 10 (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺𝑥))) = (2nd ‘(𝐹‘(𝐺𝑡))))
108107breq1d 5158 . . . . . . . . 9 (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵))
109108, 107ifbieq1d 4552 . . . . . . . 8 (𝑥 = 𝑡 → if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) = if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵))
110 fveq2 6891 . . . . . . . 8 (𝑥 = 𝑡 → (𝑥) = (𝑡))
111109, 110eleq12d 2826 . . . . . . 7 (𝑥 = 𝑡 → (if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥) ↔ if((2nd ‘(𝐹‘(𝐺𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑡))), 𝐵) ∈ (𝑡)))
112111cbvralvw 3233 . . . . . 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 712 . . . . . . . . 9 (((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) ∧ 𝑡𝑈) → (((,) ∘ 𝐹)‘(𝐺𝑡)) = 𝑡)
122 simprrl 778 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → :𝑇𝑇)
123 simprrr 779 . . . . . . . . . 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 3611 . . . . . . . . . 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 579 . . . . . . . . 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 777 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝐴𝑧)
127 simprll 776 . . . . . . . . . 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 4464 . . . . . . . . . . 11 ((𝐴𝑧𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
130126, 128, 129syl2anc 583 . . . . . . . . . 10 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)
131 ineq1 4205 . . . . . . . . . . . 12 (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵)))
132131neeq1d 2999 . . . . . . . . . . 11 (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
133132, 14elrab2 3686 . . . . . . . . . 10 (𝑧𝑇 ↔ (𝑧𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅))
134127, 130, 133sylanbrc 582 . . . . . . . . 9 ((𝜑 ∧ ((𝑧𝑈𝐴𝑧) ∧ (:𝑇𝑇 ∧ ∀𝑥𝑇 if((2nd ‘(𝐹‘(𝐺𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺𝑥))), 𝐵) ∈ (𝑥)))) → 𝑧𝑇)
135 eqid 2731 . . . . . . . . 9 seq1(( ∘ 1st ), (ℕ × {𝑧})) = seq1(( ∘ 1st ), (ℕ × {𝑧}))
136 fveq2 6891 . . . . . . . . . . 11 (𝑚 = 𝑛 → (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚) = (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛))
137136eleq2d 2818 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛)))
138137cbvrabv 3441 . . . . . . . . 9 {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1(( ∘ 1st ), (ℕ × {𝑧}))‘𝑛)}
139 eqid 2731 . . . . . . . . 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 25282 . . . . . . . 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 242 . . . . 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 1935 . . 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 3160 1 (𝜑 → (𝐵𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wex 1780  wcel 2105  wne 2939  wral 3060  wrex 3069  {crab 3431  cin 3947  wss 3948  c0 4322  ifcif 4528  𝒫 cpw 4602  {csn 4628  cop 4634   cuni 4908   class class class wbr 5148   × cxp 5674  ran crn 5677  ccom 5680  wf 6539  cfv 6543  (class class class)co 7412  1st c1st 7977  2nd c2nd 7978  Fincfn 8945  supcsup 9441  infcinf 9442  cr 11115  1c1 11117   + caddc 11119  *cxr 11254   < clt 11255  cle 11256  cmin 11451  cn 12219  (,)cioo 13331  [,]cicc 13334  seqcseq 13973  abscabs 15188
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-rp 12982  df-ioo 13335  df-ico 13337  df-icc 13338  df-fz 13492  df-fzo 13635  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640
This theorem is referenced by:  ovolicc2  25284
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