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Theorem ovoliunlem3 23212
Description: Lemma for ovoliun 23213. (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 2761 . . . 4 𝑚𝐴
2 nfcsb1v 3535 . . . 4 𝑛𝑚 / 𝑛𝐴
3 csbeq1a 3528 . . . 4 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
41, 2, 3cbviun 4530 . . 3 𝑛 ∈ ℕ 𝐴 = 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴
54fveq2i 6161 . 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 11145 . . . . . . . . 9 2 ∈ ℕ
10 nnnn0 11259 . . . . . . . . 9 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
11 nnexpcl 12829 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
129, 10, 11sylancr 694 . . . . . . . 8 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
1312nnrpd 11830 . . . . . . 7 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
14 rpdivcl 11816 . . . . . . 7 ((𝐵 ∈ ℝ+ ∧ (2↑𝑛) ∈ ℝ+) → (𝐵 / (2↑𝑛)) ∈ ℝ+)
158, 13, 14syl2an 494 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ+)
16 eqid 2621 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
1716ovolgelb 23188 . . . . . 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 1323 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
1918ralrimiva 2962 . . . 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 6643 . . . . 5 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∈ V
21 nnenom 12735 . . . . 5 ℕ ≈ ω
22 coeq2 5250 . . . . . . . . 9 (𝑓 = (𝑔𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔𝑛)))
2322rneqd 5323 . . . . . . . 8 (𝑓 = (𝑔𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔𝑛)))
2423unieqd 4419 . . . . . . 7 (𝑓 = (𝑔𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔𝑛)))
2524sseq2d 3618 . . . . . 6 (𝑓 = (𝑔𝑛) → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ (𝑔𝑛))))
26 coeq2 5250 . . . . . . . . . 10 (𝑓 = (𝑔𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔𝑛)))
2726seqeq3d 12765 . . . . . . . . 9 (𝑓 = (𝑔𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))))
2827rneqd 5323 . . . . . . . 8 (𝑓 = (𝑔𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))))
2928supeq1d 8312 . . . . . . 7 (𝑓 = (𝑔𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ))
3029breq1d 4633 . . . . . 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 746 . . . . 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 9221 . . . 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 14883 . . . . . . 7 (ℕ × ℕ) ≈ ℕ
3534ensymi 7966 . . . . . 6 ℕ ≈ (ℕ × ℕ)
36 bren 7924 . . . . . 6 (ℕ ≈ (ℕ × ℕ) ↔ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
3735, 36mpbi 220 . . . . 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 2761 . . . . . . . . . 10 𝑚(vol*‘𝐴)
41 nfcv 2761 . . . . . . . . . . 11 𝑛vol*
4241, 2nffv 6165 . . . . . . . . . 10 𝑛(vol*‘𝑚 / 𝑛𝐴)
433fveq2d 6162 . . . . . . . . . 10 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
4440, 42, 43cbvmpt 4719 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
4539, 44eqtri 2643 . . . . . . . 8 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
46 simpll 789 . . . . . . . . 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 2962 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
48 nfv 1840 . . . . . . . . . . . 12 𝑚 𝐴 ⊆ ℝ
49 nfcv 2761 . . . . . . . . . . . . 13 𝑛
502, 49nfss 3581 . . . . . . . . . . . 12 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
513sseq1d 3617 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
5248, 50, 51cbvral 3159 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5347, 52sylib 208 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5453r19.21bi 2928 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
5546, 54sylan 488 . . . . . . . 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 2962 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
5740nfel1 2775 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
5842nfel1 2775 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
5943eleq1d 2683 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
6057, 58, 59cbvral 3159 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6156, 60sylib 208 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6261r19.21bi 2928 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6346, 62sylan 488 . . . . . . . 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 761 . . . . . . . 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 761 . . . . . . . 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 2621 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚)))
68 eqid 2621 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘)))))) = seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘))))))
69 eqid 2621 . . . . . . . 8 (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘)))) = (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘))))
70 simplr 791 . . . . . . . 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 793 . . . . . . . 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 795 . . . . . . . . . . 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 1840 . . . . . . . . . . . 12 𝑚(𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
74 nfcv 2761 . . . . . . . . . . . . . 14 𝑛 ran ((,) ∘ (𝑔𝑚))
752, 74nfss 3581 . . . . . . . . . . . . 13 𝑛𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚))
76 nfcv 2761 . . . . . . . . . . . . . 14 𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < )
77 nfcv 2761 . . . . . . . . . . . . . 14 𝑛
78 nfcv 2761 . . . . . . . . . . . . . . 15 𝑛 +
79 nfcv 2761 . . . . . . . . . . . . . . 15 𝑛(𝐵 / (2↑𝑚))
8042, 78, 79nfov 6641 . . . . . . . . . . . . . 14 𝑛((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))
8176, 77, 80nfbr 4669 . . . . . . . . . . . . 13 𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))
8275, 81nfan 1825 . . . . . . . . . . . 12 𝑛(𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
83 fveq2 6158 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
8483coeq2d 5254 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((,) ∘ (𝑔𝑛)) = ((,) ∘ (𝑔𝑚)))
8584rneqd 5323 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ran ((,) ∘ (𝑔𝑛)) = ran ((,) ∘ (𝑔𝑚)))
8685unieqd 4419 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 ran ((,) ∘ (𝑔𝑛)) = ran ((,) ∘ (𝑔𝑚)))
873, 86sseq12d 3619 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐴 ran ((,) ∘ (𝑔𝑛)) ↔ 𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚))))
8883coeq2d 5254 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔𝑛)) = ((abs ∘ − ) ∘ (𝑔𝑚)))
8988seqeq3d 12765 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))))
9089rneqd 5323 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))))
9190supeq1d 8312 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ))
92 oveq2 6623 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
9392oveq2d 6631 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
9443, 93oveq12d 6633 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
9591, 94breq12d 4636 . . . . . . . . . . . . 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 746 . . . . . . . . . . . 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 3159 . . . . . . . . . . 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 208 . . . . . . . . . 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 2928 . . . . . . . . 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 475 . . . . . . . 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 479 . . . . . . . 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 23211 . . . . . . 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 629 . . . . . 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 1858 . . . . 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 1858 . . 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 4658 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2908  wrex 2909  csb 3519  cin 3559  wss 3560   cuni 4409   ciun 4492   class class class wbr 4623  cmpt 4683   × cxp 5082  ran crn 5085  ccom 5088  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  𝑚 cmap 7817  cen 7912  supcsup 8306  cr 9895  1c1 9897   + caddc 9899  *cxr 10033   < clt 10034  cle 10035  cmin 10226   / cdiv 10644  cn 10980  2c2 11030  0cn0 11252  +crp 11792  (,)cioo 12133  seqcseq 12757  cexp 12816  abscabs 13924  vol*covol 23171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cc 9217  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-ioo 12137  df-ico 12139  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-rlim 14170  df-sum 14367  df-ovol 23173
This theorem is referenced by:  ovoliun  23213
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