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Theorem ovolshftlem1 24896
Description: Lemma for ovolshft 24898. (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 24853 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
31, 2sylan 581 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
43simp1d 1143 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
53simp2d 1144 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
6 ovolshft.2 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℝ)
76adantr 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
83simp3d 1145 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
94, 5, 7, 8leadd1dd 11777 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹𝑛)) + 𝐶))
10 df-br 5110 . . . . . . . . . . 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 11192 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) + 𝐶) ∈ ℝ)
135, 7readdcld 11192 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ ℝ)
1412, 13opelxpd 5675 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ))
1511, 14elind 4158 . . . . . . . . 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 7066 . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18 eqid 2733 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
1918ovolfsf 24858 . . . . . . . 8 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
20 ffn 6672 . . . . . . . 8 (((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
2117, 19, 203syl 18 . . . . . . 7 (𝜑 → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
22 eqid 2733 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
2322ovolfsf 24858 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
24 ffn 6672 . . . . . . . 8 (((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
251, 23, 243syl 18 . . . . . . 7 (𝜑 → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
26 opex 5425 . . . . . . . . . . . . . 14 ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ V
2716fvmpt2 6963 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
2826, 27mpan2 690 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
2928fveq2d 6850 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩))
30 ovex 7394 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑛)) + 𝐶) ∈ V
31 ovex 7394 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ V
3230, 31op2nd 7934 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩) = ((2nd ‘(𝐹𝑛)) + 𝐶)
3329, 32eqtrdi 2789 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) + 𝐶))
3428fveq2d 6850 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩))
3530, 31op1st 7933 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩) = ((1st ‘(𝐹𝑛)) + 𝐶)
3634, 35eqtrdi 2789 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) + 𝐶))
3733, 36oveq12d 7379 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)))
3837adantl 483 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)))
395recnd 11191 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℂ)
404recnd 11191 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℂ)
417recnd 11191 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℂ)
4239, 40, 41pnpcan2d 11558 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4338, 42eqtrd 2773 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4418ovolfsval 24857 . . . . . . . . 9 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
4517, 44sylan 581 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
4622ovolfsval 24857 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
471, 46sylan 581 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4843, 45, 473eqtr4d 2783 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
4921, 25, 48eqfnfvd 6989 . . . . . 6 (𝜑 → ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐹))
5049seqeq3d 13923 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
51 ovolshft.5 . . . . 5 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
5250, 51eqtr4di 2791 . . . 4 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = 𝑆)
5352rneqd 5897 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) = ran 𝑆)
5453supeq1d 9390 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
55 ovolshft.3 . . . . . . . . 9 (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})
5655eleq2d 2820 . . . . . . . 8 (𝜑 → (𝑦𝐵𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴}))
57 oveq1 7368 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐶) = (𝑦𝐶))
5857eleq1d 2819 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝐶) ∈ 𝐴 ↔ (𝑦𝐶) ∈ 𝐴))
5958elrab 3649 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴))
6056, 59bitrdi 287 . . . . . . 7 (𝜑 → (𝑦𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)))
6160biimpa 478 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴))
62 breq2 5113 . . . . . . . . . 10 (𝑥 = (𝑦𝐶) → ((1st ‘(𝐹𝑛)) < 𝑥 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
63 breq1 5112 . . . . . . . . . 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 3172 . . . . . . . 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 24854 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
6967, 1, 68syl2anc 585 . . . . . . . . . 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 482 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
72 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → (𝑦𝐶) ∈ 𝐴)
7365, 71, 72rspcdva 3584 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
7436adantl 483 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) + 𝐶))
7574breq1d 5119 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ ((1st ‘(𝐹𝑛)) + 𝐶) < 𝑦))
764adantlr 714 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
776ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
78 simplrl 776 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
7976, 77, 78ltaddsubd 11763 . . . . . . . . . 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 483 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) + 𝐶))
8281breq2d 5121 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) + 𝐶)))
835adantlr 714 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
8478, 77, 83ltsubaddd 11759 . . . . . . . . . 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 3170 . . . . . . 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 592 . . . . 5 ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
9089ralrimiva 3140 . . . 4 (𝜑 → ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
91 ssrab2 4041 . . . . . 6 {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴} ⊆ ℝ
9255, 91eqsstrdi 4002 . . . . 5 (𝜑𝐵 ⊆ ℝ)
93 ovolfioo 24854 . . . . 5 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9492, 17, 93syl2anc 585 . . . 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 2733 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
9896, 97elovolmr 24863 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ran ((,) ∘ 𝐺)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
9917, 95, 98syl2anc 585 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
10054, 99eqeltrrd 2835 1 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  {crab 3406  Vcvv 3447  cin 3913  wss 3914  cop 4596   cuni 4869   class class class wbr 5109  cmpt 5192   × cxp 5635  ran crn 5638  ccom 5641   Fn wfn 6495  wf 6496  cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  m cmap 8771  supcsup 9384  cr 11058  0cc0 11059  1c1 11060   + caddc 11062  +∞cpnf 11194  *cxr 11196   < clt 11197  cle 11198  cmin 11393  cn 12161  (,)cioo 13273  [,)cico 13275  seqcseq 13915  abscabs 15128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-ioo 13277  df-ico 13279  df-fz 13434  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130
This theorem is referenced by:  ovolshftlem2  24897
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