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Theorem ovolshftlem1 24873
Description: Lemma for ovolshft 24875. (Contributed by Mario Carneiro, 22-Mar-2014.)
Hypotheses
Ref Expression
ovolshft.1 (𝜑𝐴 ⊆ ℝ)
ovolshft.2 (𝜑𝐶 ∈ ℝ)
ovolshft.3 (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})
ovolshft.4 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
ovolshft.5 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolshft.6 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
ovolshft.7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolshft.8 (𝜑𝐴 ran ((,) ∘ 𝐹))
Assertion
Ref Expression
ovolshftlem1 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
Distinct variable groups:   𝑓,𝑛,𝑥,𝑦,𝐴   𝐶,𝑓,𝑛,𝑥,𝑦   𝑛,𝐹,𝑥   𝑓,𝐺,𝑛,𝑦   𝐵,𝑓,𝑛,𝑦   𝜑,𝑓,𝑛,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑆(𝑥,𝑦,𝑓,𝑛)   𝐹(𝑦,𝑓)   𝐺(𝑥)   𝑀(𝑥,𝑦,𝑓,𝑛)

Proof of Theorem ovolshftlem1
StepHypRef Expression
1 ovolshft.7 . . . . . . . . . . . . . 14 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2 ovolfcl 24830 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
31, 2sylan 580 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
43simp1d 1142 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
53simp2d 1143 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
6 ovolshft.2 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℝ)
76adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
83simp3d 1144 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
94, 5, 7, 8leadd1dd 11769 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹𝑛)) + 𝐶))
10 df-br 5106 . . . . . . . . . . 11 (((1st ‘(𝐹𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹𝑛)) + 𝐶) ↔ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ ≤ )
119, 10sylib 217 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ ≤ )
124, 7readdcld 11184 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) + 𝐶) ∈ ℝ)
135, 7readdcld 11184 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ ℝ)
1412, 13opelxpd 5671 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ))
1511, 14elind 4154 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
16 ovolshft.6 . . . . . . . . 9 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
1715, 16fmptd 7062 . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18 eqid 2736 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
1918ovolfsf 24835 . . . . . . . 8 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
20 ffn 6668 . . . . . . . 8 (((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
2117, 19, 203syl 18 . . . . . . 7 (𝜑 → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
22 eqid 2736 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
2322ovolfsf 24835 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
24 ffn 6668 . . . . . . . 8 (((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
251, 23, 243syl 18 . . . . . . 7 (𝜑 → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
26 opex 5421 . . . . . . . . . . . . . 14 ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ V
2716fvmpt2 6959 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
2826, 27mpan2 689 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
2928fveq2d 6846 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩))
30 ovex 7390 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑛)) + 𝐶) ∈ V
31 ovex 7390 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ V
3230, 31op2nd 7930 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩) = ((2nd ‘(𝐹𝑛)) + 𝐶)
3329, 32eqtrdi 2792 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) + 𝐶))
3428fveq2d 6846 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩))
3530, 31op1st 7929 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩) = ((1st ‘(𝐹𝑛)) + 𝐶)
3634, 35eqtrdi 2792 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) + 𝐶))
3733, 36oveq12d 7375 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)))
3837adantl 482 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)))
395recnd 11183 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℂ)
404recnd 11183 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℂ)
417recnd 11183 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℂ)
4239, 40, 41pnpcan2d 11550 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4338, 42eqtrd 2776 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4418ovolfsval 24834 . . . . . . . . 9 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
4517, 44sylan 580 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
4622ovolfsval 24834 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
471, 46sylan 580 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4843, 45, 473eqtr4d 2786 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
4921, 25, 48eqfnfvd 6985 . . . . . 6 (𝜑 → ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐹))
5049seqeq3d 13914 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
51 ovolshft.5 . . . . 5 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
5250, 51eqtr4di 2794 . . . 4 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = 𝑆)
5352rneqd 5893 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) = ran 𝑆)
5453supeq1d 9382 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
55 ovolshft.3 . . . . . . . . 9 (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})
5655eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑦𝐵𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴}))
57 oveq1 7364 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐶) = (𝑦𝐶))
5857eleq1d 2822 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝐶) ∈ 𝐴 ↔ (𝑦𝐶) ∈ 𝐴))
5958elrab 3645 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴))
6056, 59bitrdi 286 . . . . . . 7 (𝜑 → (𝑦𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)))
6160biimpa 477 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴))
62 breq2 5109 . . . . . . . . . 10 (𝑥 = (𝑦𝐶) → ((1st ‘(𝐹𝑛)) < 𝑥 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
63 breq1 5108 . . . . . . . . . 10 (𝑥 = (𝑦𝐶) → (𝑥 < (2nd ‘(𝐹𝑛)) ↔ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
6462, 63anbi12d 631 . . . . . . . . 9 (𝑥 = (𝑦𝐶) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
6564rexbidv 3175 . . . . . . . 8 (𝑥 = (𝑦𝐶) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
66 ovolshft.8 . . . . . . . . . 10 (𝜑𝐴 ran ((,) ∘ 𝐹))
67 ovolshft.1 . . . . . . . . . . 11 (𝜑𝐴 ⊆ ℝ)
68 ovolfioo 24831 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
6967, 1, 68syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
7066, 69mpbid 231 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
7170adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
72 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → (𝑦𝐶) ∈ 𝐴)
7365, 71, 72rspcdva 3582 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
7436adantl 482 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) + 𝐶))
7574breq1d 5115 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ ((1st ‘(𝐹𝑛)) + 𝐶) < 𝑦))
764adantlr 713 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
776ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
78 simplrl 775 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
7976, 77, 78ltaddsubd 11755 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) + 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
8075, 79bitrd 278 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
8133adantl 482 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) + 𝐶))
8281breq2d 5117 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) + 𝐶)))
835adantlr 713 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
8478, 77, 83ltsubaddd 11751 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑦𝐶) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) + 𝐶)))
8582, 84bitr4d 281 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
8680, 85anbi12d 631 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
8786rexbidva 3173 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
8873, 87mpbird 256 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
8961, 88syldan 591 . . . . 5 ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
9089ralrimiva 3143 . . . 4 (𝜑 → ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
91 ssrab2 4037 . . . . . 6 {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴} ⊆ ℝ
9255, 91eqsstrdi 3998 . . . . 5 (𝜑𝐵 ⊆ ℝ)
93 ovolfioo 24831 . . . . 5 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9492, 17, 93syl2anc 584 . . . 4 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9590, 94mpbird 256 . . 3 (𝜑𝐵 ran ((,) ∘ 𝐺))
96 ovolshft.4 . . . 4 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
97 eqid 2736 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
9896, 97elovolmr 24840 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ran ((,) ∘ 𝐺)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
9917, 95, 98syl2anc 584 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
10054, 99eqeltrrd 2839 1 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cin 3909  wss 3910  cop 4592   cuni 4865   class class class wbr 5105  cmpt 5188   × cxp 5631  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  supcsup 9376  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cmin 11385  cn 12153  (,)cioo 13264  [,)cico 13266  seqcseq 13906  abscabs 15119
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ioo 13268  df-ico 13270  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121
This theorem is referenced by:  ovolshftlem2  24874
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