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Theorem ovoliunlem3 22993
Description: Lemma for ovoliun 22994. (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 2747 . . . 4 𝑚𝐴
2 nfcsb1v 3511 . . . 4 𝑛𝑚 / 𝑛𝐴
3 csbeq1a 3504 . . . 4 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
41, 2, 3cbviun 4484 . . 3 𝑛 ∈ ℕ 𝐴 = 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴
54fveq2i 6088 . 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 11029 . . . . . . . . 9 2 ∈ ℕ
10 nnnn0 11143 . . . . . . . . 9 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
11 nnexpcl 12687 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
129, 10, 11sylancr 693 . . . . . . . 8 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
1312nnrpd 11699 . . . . . . 7 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
14 rpdivcl 11685 . . . . . . 7 ((𝐵 ∈ ℝ+ ∧ (2↑𝑛) ∈ ℝ+) → (𝐵 / (2↑𝑛)) ∈ ℝ+)
158, 13, 14syl2an 492 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ+)
16 eqid 2606 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
1716ovolgelb 22969 . . . . . 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 1317 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
1918ralrimiva 2945 . . . 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 6552 . . . . 5 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∈ V
21 nnenom 12593 . . . . 5 ℕ ≈ ω
22 coeq2 5187 . . . . . . . . 9 (𝑓 = (𝑔𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔𝑛)))
2322rneqd 5258 . . . . . . . 8 (𝑓 = (𝑔𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔𝑛)))
2423unieqd 4373 . . . . . . 7 (𝑓 = (𝑔𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔𝑛)))
2524sseq2d 3592 . . . . . 6 (𝑓 = (𝑔𝑛) → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ (𝑔𝑛))))
26 coeq2 5187 . . . . . . . . . 10 (𝑓 = (𝑔𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔𝑛)))
2726seqeq3d 12623 . . . . . . . . 9 (𝑓 = (𝑔𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))))
2827rneqd 5258 . . . . . . . 8 (𝑓 = (𝑔𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))))
2928supeq1d 8209 . . . . . . 7 (𝑓 = (𝑔𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ))
3029breq1d 4584 . . . . . 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 742 . . . . 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 9118 . . . 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 14721 . . . . . . 7 (ℕ × ℕ) ≈ ℕ
3534ensymi 7866 . . . . . 6 ℕ ≈ (ℕ × ℕ)
36 bren 7824 . . . . . 6 (ℕ ≈ (ℕ × ℕ) ↔ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
3735, 36mpbi 218 . . . . 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 2747 . . . . . . . . . 10 𝑚(vol*‘𝐴)
41 nfcv 2747 . . . . . . . . . . 11 𝑛vol*
4241, 2nffv 6092 . . . . . . . . . 10 𝑛(vol*‘𝑚 / 𝑛𝐴)
433fveq2d 6089 . . . . . . . . . 10 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
4440, 42, 43cbvmpt 4668 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
4539, 44eqtri 2628 . . . . . . . 8 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
46 simpll 785 . . . . . . . . 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)))))) → 𝜑)
476ralrimiva 2945 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
48 nfv 1829 . . . . . . . . . . . 12 𝑚 𝐴 ⊆ ℝ
49 nfcv 2747 . . . . . . . . . . . . 13 𝑛
502, 49nfss 3557 . . . . . . . . . . . 12 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
513sseq1d 3591 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
5248, 50, 51cbvral 3139 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5347, 52sylib 206 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5453r19.21bi 2912 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
5546, 54sylan 486 . . . . . . . 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)
567ralrimiva 2945 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
5740nfel1 2761 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
5842nfel1 2761 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
5943eleq1d 2668 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
6057, 58, 59cbvral 3139 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6156, 60sylib 206 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6261r19.21bi 2912 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6346, 62sylan 486 . . . . . . . 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)
64 ovoliun.r . . . . . . . . 9 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
6564ad2antrr 757 . . . . . . . 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)
668ad2antrr 757 . . . . . . . 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+)
67 eqid 2606 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚)))
68 eqid 2606 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘)))))) = seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘))))))
69 eqid 2606 . . . . . . . 8 (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘)))) = (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘))))
70 simplr 787 . . . . . . . 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))
71 simprl 789 . . . . . . . 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))
72 simprr 791 . . . . . . . . . . 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)))))
73 nfv 1829 . . . . . . . . . . . 12 𝑚(𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
74 nfcv 2747 . . . . . . . . . . . . . 14 𝑛 ran ((,) ∘ (𝑔𝑚))
752, 74nfss 3557 . . . . . . . . . . . . 13 𝑛𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚))
76 nfcv 2747 . . . . . . . . . . . . . 14 𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < )
77 nfcv 2747 . . . . . . . . . . . . . 14 𝑛
78 nfcv 2747 . . . . . . . . . . . . . . 15 𝑛 +
79 nfcv 2747 . . . . . . . . . . . . . . 15 𝑛(𝐵 / (2↑𝑚))
8042, 78, 79nfov 6550 . . . . . . . . . . . . . 14 𝑛((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))
8176, 77, 80nfbr 4620 . . . . . . . . . . . . 13 𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))
8275, 81nfan 1815 . . . . . . . . . . . 12 𝑛(𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
83 fveq2 6085 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
8483coeq2d 5191 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((,) ∘ (𝑔𝑛)) = ((,) ∘ (𝑔𝑚)))
8584rneqd 5258 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ran ((,) ∘ (𝑔𝑛)) = ran ((,) ∘ (𝑔𝑚)))
8685unieqd 4373 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 ran ((,) ∘ (𝑔𝑛)) = ran ((,) ∘ (𝑔𝑚)))
873, 86sseq12d 3593 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐴 ran ((,) ∘ (𝑔𝑛)) ↔ 𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚))))
8883coeq2d 5191 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔𝑛)) = ((abs ∘ − ) ∘ (𝑔𝑚)))
8988seqeq3d 12623 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))))
9089rneqd 5258 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))))
9190supeq1d 8209 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ))
92 oveq2 6532 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
9392oveq2d 6540 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
9443, 93oveq12d 6542 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
9591, 94breq12d 4587 . . . . . . . . . . . . 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)))))
9687, 95anbi12d 742 . . . . . . . . . . . 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))))))
9773, 82, 96cbvral 3139 . . . . . . . . . . 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)))))
9872, 97sylib 206 . . . . . . . . . 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)))))
9998r19.21bi 2912 . . . . . . . . 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)))))
10099simpld 473 . . . . . . . 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)))
10199simprd 477 . . . . . . . 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))))
10238, 45, 55, 63, 65, 66, 67, 68, 69, 70, 71, 100, 101ovoliunlem2 22992 . . . . . . 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))
103102exp31 627 . . . . . 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))))
104103exlimdv 1847 . . . . 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))))
10537, 104mpi 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)))
106105exlimdv 1847 . . 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)))
10733, 106mpd 15 . 2 (𝜑 → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
1085, 107syl5eqbr 4609 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1976  wral 2892  wrex 2893  csb 3495  cin 3535  wss 3536   cuni 4363   ciun 4446   class class class wbr 4574  cmpt 4634   × cxp 5023  ran crn 5026  ccom 5029  wf 5783  1-1-ontowf1o 5786  cfv 5787  (class class class)co 6524  1st c1st 7031  2nd c2nd 7032  𝑚 cmap 7718  cen 7812  supcsup 8203  cr 9788  1c1 9790   + caddc 9792  *cxr 9926   < clt 9927  cle 9928  cmin 10114   / cdiv 10530  cn 10864  2c2 10914  0cn0 11136  +crp 11661  (,)cioo 11999  seqcseq 12615  cexp 12674  abscabs 13765  vol*covol 22952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cc 9114  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-inf 8206  df-oi 8272  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-ioo 12003  df-ico 12005  df-fz 12150  df-fzo 12287  df-fl 12407  df-seq 12616  df-exp 12675  df-hash 12932  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-clim 14010  df-rlim 14011  df-sum 14208  df-ovol 22954
This theorem is referenced by:  ovoliun  22994
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