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Theorem ovolshftlem1 23323
 Description: Lemma for ovolshft 23325. (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 23281 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
31, 2sylan 487 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
43simp1d 1093 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
53simp2d 1094 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
6 ovolshft.2 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ ℝ)
76adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
83simp3d 1095 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
94, 5, 7, 8leadd1dd 10679 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹𝑛)) + 𝐶))
10 df-br 4686 . . . . . . . . . . 11 (((1st ‘(𝐹𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹𝑛)) + 𝐶) ↔ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ ≤ )
119, 10sylib 208 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ ≤ )
124, 7readdcld 10107 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) + 𝐶) ∈ ℝ)
135, 7readdcld 10107 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ ℝ)
14 opelxp 5180 . . . . . . . . . . 11 (⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ) ↔ (((1st ‘(𝐹𝑛)) + 𝐶) ∈ ℝ ∧ ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ ℝ))
1512, 13, 14sylanbrc 699 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ))
1611, 15elind 3831 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
17 ovolshft.6 . . . . . . . . 9 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
1816, 17fmptd 6425 . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19 eqid 2651 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
2019ovolfsf 23286 . . . . . . . 8 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
21 ffn 6083 . . . . . . . 8 (((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
2218, 20, 213syl 18 . . . . . . 7 (𝜑 → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
23 eqid 2651 . . . . . . . . 9 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
2423ovolfsf 23286 . . . . . . . 8 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
25 ffn 6083 . . . . . . . 8 (((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
261, 24, 253syl 18 . . . . . . 7 (𝜑 → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
27 opex 4962 . . . . . . . . . . . . . 14 ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ V
2817fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
2927, 28mpan2 707 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩)
3029fveq2d 6233 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩))
31 ovex 6718 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑛)) + 𝐶) ∈ V
32 ovex 6718 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑛)) + 𝐶) ∈ V
3331, 32op2nd 7219 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩) = ((2nd ‘(𝐹𝑛)) + 𝐶)
3430, 33syl6eq 2701 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) + 𝐶))
3529fveq2d 6233 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩))
3631, 32op1st 7218 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑛)) + 𝐶), ((2nd ‘(𝐹𝑛)) + 𝐶)⟩) = ((1st ‘(𝐹𝑛)) + 𝐶)
3735, 36syl6eq 2701 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) + 𝐶))
3834, 37oveq12d 6708 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)))
3938adantl 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)))
405recnd 10106 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℂ)
414recnd 10106 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℂ)
427recnd 10106 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐶 ∈ ℂ)
4340, 41, 42pnpcan2d 10468 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((2nd ‘(𝐹𝑛)) + 𝐶) − ((1st ‘(𝐹𝑛)) + 𝐶)) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4439, 43eqtrd 2685 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4519ovolfsval 23285 . . . . . . . . 9 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
4618, 45sylan 487 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
4723ovolfsval 23285 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
481, 47sylan 487 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
4944, 46, 483eqtr4d 2695 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
5022, 26, 49eqfnfvd 6354 . . . . . 6 (𝜑 → ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐹))
5150seqeq3d 12849 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
52 ovolshft.5 . . . . 5 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
5351, 52syl6eqr 2703 . . . 4 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = 𝑆)
5453rneqd 5385 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) = ran 𝑆)
5554supeq1d 8393 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
56 ovolshft.3 . . . . . . . . 9 (𝜑𝐵 = {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴})
5756eleq2d 2716 . . . . . . . 8 (𝜑 → (𝑦𝐵𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴}))
58 oveq1 6697 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐶) = (𝑦𝐶))
5958eleq1d 2715 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝐶) ∈ 𝐴 ↔ (𝑦𝐶) ∈ 𝐴))
6059elrab 3396 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴))
6157, 60syl6bb 276 . . . . . . 7 (𝜑 → (𝑦𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)))
6261biimpa 500 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴))
63 simprr 811 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → (𝑦𝐶) ∈ 𝐴)
64 ovolshft.8 . . . . . . . . . 10 (𝜑𝐴 ran ((,) ∘ 𝐹))
65 ovolshft.1 . . . . . . . . . . 11 (𝜑𝐴 ⊆ ℝ)
66 ovolfioo 23282 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
6765, 1, 66syl2anc 694 . . . . . . . . . 10 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛)))))
6864, 67mpbid 222 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
6968adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))))
70 breq2 4689 . . . . . . . . . . 11 (𝑥 = (𝑦𝐶) → ((1st ‘(𝐹𝑛)) < 𝑥 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
71 breq1 4688 . . . . . . . . . . 11 (𝑥 = (𝑦𝐶) → (𝑥 < (2nd ‘(𝐹𝑛)) ↔ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
7270, 71anbi12d 747 . . . . . . . . . 10 (𝑥 = (𝑦𝐶) → (((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
7372rexbidv 3081 . . . . . . . . 9 (𝑥 = (𝑦𝐶) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
7473rspcv 3336 . . . . . . . 8 ((𝑦𝐶) ∈ 𝐴 → (∀𝑥𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < 𝑥𝑥 < (2nd ‘(𝐹𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
7563, 69, 74sylc 65 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
7637adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) + 𝐶))
7776breq1d 4695 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ ((1st ‘(𝐹𝑛)) + 𝐶) < 𝑦))
784adantlr 751 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
796ad2antrr 762 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
80 simplrl 817 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
8178, 79, 80ltaddsubd 10665 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹𝑛)) + 𝐶) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
8277, 81bitrd 268 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < 𝑦 ↔ (1st ‘(𝐹𝑛)) < (𝑦𝐶)))
8334adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = ((2nd ‘(𝐹𝑛)) + 𝐶))
8483breq2d 4697 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) + 𝐶)))
855adantlr 751 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
8680, 79, 85ltsubaddd 10661 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑦𝐶) < (2nd ‘(𝐹𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹𝑛)) + 𝐶)))
8784, 86bitr4d 271 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺𝑛)) ↔ (𝑦𝐶) < (2nd ‘(𝐹𝑛))))
8882, 87anbi12d 747 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))) ↔ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
8988rexbidva 3078 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) < (𝑦𝐶) ∧ (𝑦𝐶) < (2nd ‘(𝐹𝑛)))))
9075, 89mpbird 247 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
9162, 90syldan 486 . . . . 5 ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
9291ralrimiva 2995 . . . 4 (𝜑 → ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛))))
93 ssrab2 3720 . . . . . 6 {𝑥 ∈ ℝ ∣ (𝑥𝐶) ∈ 𝐴} ⊆ ℝ
9456, 93syl6eqss 3688 . . . . 5 (𝜑𝐵 ⊆ ℝ)
95 ovolfioo 23282 . . . . 5 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9694, 18, 95syl2anc 694 . . . 4 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑦𝐵𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) < 𝑦𝑦 < (2nd ‘(𝐺𝑛)))))
9792, 96mpbird 247 . . 3 (𝜑𝐵 ran ((,) ∘ 𝐺))
98 ovolshft.4 . . . 4 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
99 eqid 2651 . . . 4 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
10098, 99elovolmr 23290 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ran ((,) ∘ 𝐺)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
10118, 97, 100syl2anc 694 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
10255, 101eqeltrrd 2731 1 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  ⟨cop 4216  ∪ cuni 4468   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  ran crn 5144   ∘ ccom 5147   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209   ↑𝑚 cmap 7899  supcsup 8387  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112   ≤ cle 10113   − cmin 10304  ℕcn 11058  (,)cioo 12213  [,)cico 12215  seqcseq 12841  abscabs 14018 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ioo 12217  df-ico 12219  df-fz 12365  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020 This theorem is referenced by:  ovolshftlem2  23324
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