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Theorem ovolshftlem1 25557
Description: Lemma for ovolshft 25559. (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 25514 . . . . . . . . . . . . . 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 1141 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
53simp2d 1142 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
6 ovolshft.2 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℝ)
76adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
83simp3d 1143 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
94, 5, 7, 8leadd1dd 11874 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹𝑛)) + 𝐶))
10 df-br 5148 . . . . . . . . . . 11 (((1st ‘(𝐹𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹𝑛)) + 𝐶) ↔ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ ≤ )
119, 10sylib 218 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ ≤ )
124, 7readdcld 11287 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) + 𝐶) ∈ ℝ)
135, 7readdcld 11287 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ ℝ)
1412, 13opelxpd 5727 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ))
1511, 14elind 4209 . . . . . . . . 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 7133 . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18 eqid 2734 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
1918ovolfsf 25519 . . . . . . . 8 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
20 ffn 6736 . . . . . . . 8 (((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
2117, 19, 203syl 18 . . . . . . 7 (𝜑 → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
22 eqid 2734 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
2322ovolfsf 25519 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
24 ffn 6736 . . . . . . . 8 (((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
251, 23, 243syl 18 . . . . . . 7 (𝜑 → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
26 opex 5474 . . . . . . . . . . . . . 14 ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ V
2716fvmpt2 7026 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
2826, 27mpan2 691 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
2928fveq2d 6910 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩))
30 ovex 7463 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑛)) + 𝐶) ∈ V
31 ovex 7463 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ V
3230, 31op2nd 8021 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩) = ((2nd ‘(𝐹𝑛)) + 𝐶)
3329, 32eqtrdi 2790 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) + 𝐶))
3428fveq2d 6910 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩))
3530, 31op1st 8020 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩) = ((1st ‘(𝐹𝑛)) + 𝐶)
3634, 35eqtrdi 2790 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) + 𝐶))
3733, 36oveq12d 7448 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)))
3837adantl 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)))
395recnd 11286 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℂ)
404recnd 11286 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℂ)
417recnd 11286 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℂ)
4239, 40, 41pnpcan2d 11655 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4338, 42eqtrd 2774 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4418ovolfsval 25518 . . . . . . . . 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 25518 . . . . . . . . 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 2784 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
4921, 25, 48eqfnfvd 7053 . . . . . 6 (𝜑 → ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐹))
5049seqeq3d 14046 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
51 ovolshft.5 . . . . 5 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
5250, 51eqtr4di 2792 . . . 4 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = 𝑆)
5352rneqd 5951 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) = ran 𝑆)
5453supeq1d 9483 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
55 ovolshft.3 . . . . . . . . 9 (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})
5655eleq2d 2824 . . . . . . . 8 (𝜑 → (𝑦𝐵𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴}))
57 oveq1 7437 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐶) = (𝑦𝐶))
5857eleq1d 2823 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝐶) ∈ 𝐴 ↔ (𝑦𝐶) ∈ 𝐴))
5958elrab 3694 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴))
6056, 59bitrdi 287 . . . . . . 7 (𝜑 → (𝑦𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)))
6160biimpa 476 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴))
62 breq2 5151 . . . . . . . . . 10 (𝑥 = (𝑦𝐶) → ((1st ‘(𝐹𝑛)) < 𝑥 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
63 breq1 5150 . . . . . . . . . 10 (𝑥 = (𝑦𝐶) → (𝑥 < (2nd ‘(𝐹𝑛)) ↔ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
6462, 63anbi12d 632 . . . . . . . . 9 (𝑥 = (𝑦𝐶) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
6564rexbidv 3176 . . . . . . . 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 25515 . . . . . . . . . . 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 232 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
7170adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
72 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → (𝑦𝐶) ∈ 𝐴)
7365, 71, 72rspcdva 3622 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
7436adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) + 𝐶))
7574breq1d 5157 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ ((1st ‘(𝐹𝑛)) + 𝐶) < 𝑦))
764adantlr 715 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
776ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
78 simplrl 777 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
7976, 77, 78ltaddsubd 11860 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) + 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
8075, 79bitrd 279 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
8133adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) + 𝐶))
8281breq2d 5159 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) + 𝐶)))
835adantlr 715 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
8478, 77, 83ltsubaddd 11856 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑦𝐶) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) + 𝐶)))
8582, 84bitr4d 282 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
8680, 85anbi12d 632 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
8786rexbidva 3174 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
8873, 87mpbird 257 . . . . . 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 4089 . . . . . 6 {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴} ⊆ ℝ
9255, 91eqsstrdi 4049 . . . . 5 (𝜑𝐵 ⊆ ℝ)
93 ovolfioo 25515 . . . . 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 257 . . 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 2734 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
9896, 97elovolmr 25524 . . 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cin 3961  wss 3962  cop 4636   cuni 4911   class class class wbr 5147  cmpt 5230   × cxp 5686  ran crn 5689  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  m cmap 8864  supcsup 9477  cr 11151  0cc0 11152  1c1 11153   + caddc 11155  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293  cmin 11489  cn 12263  (,)cioo 13383  [,)cico 13385  seqcseq 14038  abscabs 15269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-ioo 13387  df-ico 13389  df-fz 13544  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271
This theorem is referenced by:  ovolshftlem2  25558
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