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Theorem ovoliunlem3 25552
Description: Lemma for ovoliun 25553. (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovoliun.t 𝑇 = seq1( + , 𝐺)
ovoliun.g 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
ovoliun.r (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
ovoliun.b (𝜑𝐵 ∈ ℝ+)
Assertion
Ref Expression
ovoliunlem3 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Distinct variable groups:   𝐵,𝑛   𝜑,𝑛   𝑛,𝐺   𝑇,𝑛
Allowed substitution hint:   𝐴(𝑛)

Proof of Theorem ovoliunlem3
Dummy variables 𝑓 𝑔 𝑗 𝑘 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2902 . . . 4 𝑚𝐴
2 nfcsb1v 3932 . . . 4 𝑛𝑚 / 𝑛𝐴
3 csbeq1a 3921 . . . 4 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
41, 2, 3cbviun 5040 . . 3 𝑛 ∈ ℕ 𝐴 = 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴
54fveq2i 6909 . 2 (vol*‘ 𝑛 ∈ ℕ 𝐴) = (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴)
6 ovoliun.a . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
7 ovoliun.v . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
8 ovoliun.b . . . . . . 7 (𝜑𝐵 ∈ ℝ+)
9 2nn 12336 . . . . . . . . 9 2 ∈ ℕ
10 nnnn0 12530 . . . . . . . . 9 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
11 nnexpcl 14111 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
129, 10, 11sylancr 587 . . . . . . . 8 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
1312nnrpd 13072 . . . . . . 7 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
14 rpdivcl 13057 . . . . . . 7 ((𝐵 ∈ ℝ+ ∧ (2↑𝑛) ∈ ℝ+) → (𝐵 / (2↑𝑛)) ∈ ℝ+)
158, 13, 14syl2an 596 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ+)
16 eqid 2734 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
1716ovolgelb 25528 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ (𝐵 / (2↑𝑛)) ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
186, 7, 15, 17syl3anc 1370 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
1918ralrimiva 3143 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
20 ovex 7463 . . . . 5 (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∈ V
21 nnenom 14017 . . . . 5 ℕ ≈ ω
22 coeq2 5871 . . . . . . . . 9 (𝑓 = (𝑔𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔𝑛)))
2322rneqd 5951 . . . . . . . 8 (𝑓 = (𝑔𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔𝑛)))
2423unieqd 4924 . . . . . . 7 (𝑓 = (𝑔𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔𝑛)))
2524sseq2d 4027 . . . . . 6 (𝑓 = (𝑔𝑛) → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ (𝑔𝑛))))
26 coeq2 5871 . . . . . . . . . 10 (𝑓 = (𝑔𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔𝑛)))
2726seqeq3d 14046 . . . . . . . . 9 (𝑓 = (𝑔𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))))
2827rneqd 5951 . . . . . . . 8 (𝑓 = (𝑔𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))))
2928supeq1d 9483 . . . . . . 7 (𝑓 = (𝑔𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ))
3029breq1d 5157 . . . . . 6 (𝑓 = (𝑔𝑛) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ↔ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
3125, 30anbi12d 632 . . . . 5 (𝑓 = (𝑔𝑛) → ((𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))) ↔ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))))
3220, 21, 31axcc4 10476 . . . 4 (∀𝑛 ∈ ℕ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))) → ∃𝑔(𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))))
3319, 32syl 17 . . 3 (𝜑 → ∃𝑔(𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))))
34 xpnnen 16243 . . . . . . 7 (ℕ × ℕ) ≈ ℕ
3534ensymi 9042 . . . . . 6 ℕ ≈ (ℕ × ℕ)
36 bren 8993 . . . . . 6 (ℕ ≈ (ℕ × ℕ) ↔ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
3735, 36mpbi 230 . . . . 5 𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ)
38 ovoliun.t . . . . . . . 8 𝑇 = seq1( + , 𝐺)
39 ovoliun.g . . . . . . . . 9 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
40 nfcv 2902 . . . . . . . . . 10 𝑚(vol*‘𝐴)
41 nfcv 2902 . . . . . . . . . . 11 𝑛vol*
4241, 2nffv 6916 . . . . . . . . . 10 𝑛(vol*‘𝑚 / 𝑛𝐴)
433fveq2d 6910 . . . . . . . . . 10 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
4440, 42, 43cbvmpt 5258 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
4539, 44eqtri 2762 . . . . . . . 8 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
466ralrimiva 3143 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
47 nfv 1911 . . . . . . . . . . . 12 𝑚 𝐴 ⊆ ℝ
48 nfcv 2902 . . . . . . . . . . . . 13 𝑛
492, 48nfss 3987 . . . . . . . . . . . 12 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
503sseq1d 4026 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
5147, 49, 50cbvralw 3303 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5246, 51sylib 218 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5352r19.21bi 3248 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
5453ad4ant14 752 . . . . . . . 8 ((((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
557ralrimiva 3143 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
5640nfel1 2919 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
5742nfel1 2919 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
5843eleq1d 2823 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
5956, 57, 58cbvralw 3303 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6055, 59sylib 218 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6160r19.21bi 3248 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6261ad4ant14 752 . . . . . . . 8 ((((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
63 ovoliun.r . . . . . . . . 9 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
6463ad2antrr 726 . . . . . . . 8 (((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
658ad2antrr 726 . . . . . . . 8 (((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) → 𝐵 ∈ ℝ+)
66 eqid 2734 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚)))
67 eqid 2734 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘)))))) = seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘))))))
68 eqid 2734 . . . . . . . 8 (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘)))) = (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘))))
69 simplr 769 . . . . . . . 8 (((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) → 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
70 simprl 771 . . . . . . . 8 (((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) → 𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
71 simprr 773 . . . . . . . . . . 11 (((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) → ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
72 nfv 1911 . . . . . . . . . . . 12 𝑚(𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
73 nfcv 2902 . . . . . . . . . . . . . 14 𝑛 ran ((,) ∘ (𝑔𝑚))
742, 73nfss 3987 . . . . . . . . . . . . 13 𝑛𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚))
75 nfcv 2902 . . . . . . . . . . . . . 14 𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < )
76 nfcv 2902 . . . . . . . . . . . . . 14 𝑛
77 nfcv 2902 . . . . . . . . . . . . . . 15 𝑛 +
78 nfcv 2902 . . . . . . . . . . . . . . 15 𝑛(𝐵 / (2↑𝑚))
7942, 77, 78nfov 7460 . . . . . . . . . . . . . 14 𝑛((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))
8075, 76, 79nfbr 5194 . . . . . . . . . . . . 13 𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))
8174, 80nfan 1896 . . . . . . . . . . . 12 𝑛(𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
82 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
8382coeq2d 5875 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((,) ∘ (𝑔𝑛)) = ((,) ∘ (𝑔𝑚)))
8483rneqd 5951 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ran ((,) ∘ (𝑔𝑛)) = ran ((,) ∘ (𝑔𝑚)))
8584unieqd 4924 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 ran ((,) ∘ (𝑔𝑛)) = ran ((,) ∘ (𝑔𝑚)))
863, 85sseq12d 4028 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐴 ran ((,) ∘ (𝑔𝑛)) ↔ 𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚))))
8782coeq2d 5875 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔𝑛)) = ((abs ∘ − ) ∘ (𝑔𝑚)))
8887seqeq3d 14046 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))))
8988rneqd 5951 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))))
9089supeq1d 9483 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ))
91 oveq2 7438 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
9291oveq2d 7446 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
9343, 92oveq12d 7448 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
9490, 93breq12d 5160 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ↔ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))))
9586, 94anbi12d 632 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))) ↔ (𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))))
9672, 81, 95cbvralw 3303 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))) ↔ ∀𝑚 ∈ ℕ (𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))))
9771, 96sylib 218 . . . . . . . . . 10 (((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) → ∀𝑚 ∈ ℕ (𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))))
9897r19.21bi 3248 . . . . . . . . 9 ((((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → (𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))))
9998simpld 494 . . . . . . . 8 ((((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)))
10098simprd 495 . . . . . . . 8 ((((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
10138, 45, 54, 62, 64, 65, 66, 67, 68, 69, 70, 99, 100ovoliunlem2 25551 . . . . . . 7 (((𝜑𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
102101exp31 419 . . . . . 6 (𝜑 → (𝑗:ℕ–1-1-onto→(ℕ × ℕ) → ((𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))))
103102exlimdv 1930 . . . . 5 (𝜑 → (∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ) → ((𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))))
10437, 103mpi 20 . . . 4 (𝜑 → ((𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
105104exlimdv 1930 . . 3 (𝜑 → (∃𝑔(𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
10633, 105mpd 15 . 2 (𝜑 → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
1075, 106eqbrtrid 5182 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wex 1775  wcel 2105  wral 3058  wrex 3067  csb 3907  cin 3961  wss 3962   cuni 4911   ciun 4995   class class class wbr 5147  cmpt 5230   × cxp 5686  ran crn 5689  ccom 5692  wf 6558  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  m cmap 8864  cen 8980  supcsup 9477  cr 11151  1c1 11153   + caddc 11155  *cxr 11291   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  cn 12263  2c2 12318  0cn0 12523  +crp 13031  (,)cioo 13383  seqcseq 14038  cexp 14098  abscabs 15269  vol*covol 25510
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-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  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-int 4951  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-se 5641  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-isom 6571  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-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  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-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-ovol 25512
This theorem is referenced by:  ovoliun  25553
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