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Theorem ovoliunlem3 25021
Description: Lemma for ovoliun 25022. (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 2904 . . . 4 𝑚𝐴
2 nfcsb1v 3919 . . . 4 𝑛𝑚 / 𝑛𝐴
3 csbeq1a 3908 . . . 4 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
41, 2, 3cbviun 5040 . . 3 𝑛 ∈ ℕ 𝐴 = 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴
54fveq2i 6895 . 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 12285 . . . . . . . . 9 2 ∈ ℕ
10 nnnn0 12479 . . . . . . . . 9 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
11 nnexpcl 14040 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
129, 10, 11sylancr 588 . . . . . . . 8 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
1312nnrpd 13014 . . . . . . 7 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
14 rpdivcl 12999 . . . . . . 7 ((𝐵 ∈ ℝ+ ∧ (2↑𝑛) ∈ ℝ+) → (𝐵 / (2↑𝑛)) ∈ ℝ+)
158, 13, 14syl2an 597 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ+)
16 eqid 2733 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
1716ovolgelb 24997 . . . . . 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 1372 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
1918ralrimiva 3147 . . . 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 7442 . . . . 5 (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∈ V
21 nnenom 13945 . . . . 5 ℕ ≈ ω
22 coeq2 5859 . . . . . . . . 9 (𝑓 = (𝑔𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔𝑛)))
2322rneqd 5938 . . . . . . . 8 (𝑓 = (𝑔𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔𝑛)))
2423unieqd 4923 . . . . . . 7 (𝑓 = (𝑔𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔𝑛)))
2524sseq2d 4015 . . . . . 6 (𝑓 = (𝑔𝑛) → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ (𝑔𝑛))))
26 coeq2 5859 . . . . . . . . . 10 (𝑓 = (𝑔𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔𝑛)))
2726seqeq3d 13974 . . . . . . . . 9 (𝑓 = (𝑔𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))))
2827rneqd 5938 . . . . . . . 8 (𝑓 = (𝑔𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))))
2928supeq1d 9441 . . . . . . 7 (𝑓 = (𝑔𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ))
3029breq1d 5159 . . . . . 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 10434 . . . 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 16154 . . . . . . 7 (ℕ × ℕ) ≈ ℕ
3534ensymi 9000 . . . . . 6 ℕ ≈ (ℕ × ℕ)
36 bren 8949 . . . . . 6 (ℕ ≈ (ℕ × ℕ) ↔ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
3735, 36mpbi 229 . . . . 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 2904 . . . . . . . . . 10 𝑚(vol*‘𝐴)
41 nfcv 2904 . . . . . . . . . . 11 𝑛vol*
4241, 2nffv 6902 . . . . . . . . . 10 𝑛(vol*‘𝑚 / 𝑛𝐴)
433fveq2d 6896 . . . . . . . . . 10 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
4440, 42, 43cbvmpt 5260 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
4539, 44eqtri 2761 . . . . . . . 8 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
466ralrimiva 3147 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
47 nfv 1918 . . . . . . . . . . . 12 𝑚 𝐴 ⊆ ℝ
48 nfcv 2904 . . . . . . . . . . . . 13 𝑛
492, 48nfss 3975 . . . . . . . . . . . 12 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
503sseq1d 4014 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
5147, 49, 50cbvralw 3304 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5246, 51sylib 217 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5352r19.21bi 3249 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
5453ad4ant14 751 . . . . . . . 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 3147 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
5640nfel1 2920 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
5742nfel1 2920 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
5843eleq1d 2819 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
5956, 57, 58cbvralw 3304 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6055, 59sylib 217 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6160r19.21bi 3249 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6261ad4ant14 751 . . . . . . . 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 725 . . . . . . . 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 725 . . . . . . . 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 2733 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚)))
67 eqid 2733 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘)))))) = seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘))))))
68 eqid 2733 . . . . . . . 8 (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘)))) = (𝑘 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑗𝑘)))‘(2nd ‘(𝑗𝑘))))
69 simplr 768 . . . . . . . 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 770 . . . . . . . 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 772 . . . . . . . . . . 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 1918 . . . . . . . . . . . 12 𝑚(𝐴 ran ((,) ∘ (𝑔𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
73 nfcv 2904 . . . . . . . . . . . . . 14 𝑛 ran ((,) ∘ (𝑔𝑚))
742, 73nfss 3975 . . . . . . . . . . . . 13 𝑛𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚))
75 nfcv 2904 . . . . . . . . . . . . . 14 𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < )
76 nfcv 2904 . . . . . . . . . . . . . 14 𝑛
77 nfcv 2904 . . . . . . . . . . . . . . 15 𝑛 +
78 nfcv 2904 . . . . . . . . . . . . . . 15 𝑛(𝐵 / (2↑𝑚))
7942, 77, 78nfov 7439 . . . . . . . . . . . . . 14 𝑛((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))
8075, 76, 79nfbr 5196 . . . . . . . . . . . . 13 𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚)))
8174, 80nfan 1903 . . . . . . . . . . . 12 𝑛(𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ) ≤ ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
82 fveq2 6892 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
8382coeq2d 5863 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((,) ∘ (𝑔𝑛)) = ((,) ∘ (𝑔𝑚)))
8483rneqd 5938 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ran ((,) ∘ (𝑔𝑛)) = ran ((,) ∘ (𝑔𝑚)))
8584unieqd 4923 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 ran ((,) ∘ (𝑔𝑛)) = ran ((,) ∘ (𝑔𝑚)))
863, 85sseq12d 4016 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐴 ran ((,) ∘ (𝑔𝑛)) ↔ 𝑚 / 𝑛𝐴 ran ((,) ∘ (𝑔𝑚))))
8782coeq2d 5863 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔𝑛)) = ((abs ∘ − ) ∘ (𝑔𝑚)))
8887seqeq3d 13974 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))))
8988rneqd 5938 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))))
9089supeq1d 9441 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑛))), ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔𝑚))), ℝ*, < ))
91 oveq2 7417 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
9291oveq2d 7425 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
9343, 92oveq12d 7427 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol*‘𝑚 / 𝑛𝐴) + (𝐵 / (2↑𝑚))))
9490, 93breq12d 5162 . . . . . . . . . . . . 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 3304 . . . . . . . . . . 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 217 . . . . . . . . . 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 3249 . . . . . . . . 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 496 . . . . . . . 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 497 . . . . . . . 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 25020 . . . . . . 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 421 . . . . . 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 1937 . . . . 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 1937 . . 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 5184 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3062  wrex 3071  csb 3894  cin 3948  wss 3949   cuni 4909   ciun 4998   class class class wbr 5149  cmpt 5232   × cxp 5675  ran crn 5678  ccom 5681  wf 6540  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  m cmap 8820  cen 8936  supcsup 9435  cr 11109  1c1 11111   + caddc 11113  *cxr 11247   < clt 11248  cle 11249  cmin 11444   / cdiv 11871  cn 12212  2c2 12267  0cn0 12472  +crp 12974  (,)cioo 13324  seqcseq 13966  cexp 14027  abscabs 15181  vol*covol 24979
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cc 10430  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-ioo 13328  df-ico 13330  df-fz 13485  df-fzo 13628  df-fl 13757  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-sum 15633  df-ovol 24981
This theorem is referenced by:  ovoliun  25022
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