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Theorem ovollb2lem 25004
Description: Lemma for ovollb2 25005. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
ovollb2.1 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovollb2.2 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)
ovollb2.3 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ovollb2.4 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovollb2.5 (𝜑𝐴 ran ([,] ∘ 𝐹))
ovollb2.6 (𝜑𝐵 ∈ ℝ+)
ovollb2.7 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
Assertion
Ref Expression
ovollb2lem (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
Distinct variable groups:   𝐴,𝑛   𝑛,𝐹   𝐵,𝑛   𝜑,𝑛   𝑆,𝑛
Allowed substitution hints:   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovollb2lem
Dummy variables 𝑚 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovollb2.5 . . . 4 (𝜑𝐴 ran ([,] ∘ 𝐹))
2 ovollb2.4 . . . . 5 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 ovolficcss 24985 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
42, 3syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
51, 4sstrd 3992 . . 3 (𝜑𝐴 ⊆ ℝ)
6 ovolcl 24994 . . 3 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
75, 6syl 17 . 2 (𝜑 → (vol*‘𝐴) ∈ ℝ*)
8 ovolfcl 24982 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
92, 8sylan 580 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
109simp1d 1142 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
11 ovollb2.6 . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ ℝ+)
1211rphalfcld 13027 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 / 2) ∈ ℝ+)
1312adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
14 2nn 12284 . . . . . . . . . . . . . . 15 2 ∈ ℕ
15 nnnn0 12478 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1615adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17 nnexpcl 14039 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1814, 16, 17sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℕ)
1918nnrpd 13013 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
2013, 19rpdivcld 13032 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ+)
2120rpred 13015 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ)
2210, 21resubcld 11641 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
239simp2d 1143 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2423, 21readdcld 11242 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
2510, 20ltsubrpd 13047 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < (1st ‘(𝐹𝑛)))
269simp3d 1144 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
2723, 20ltaddrpd 13048 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2810, 23, 24, 26, 27lelttrd 11371 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2922, 10, 24, 25, 28lttrd 11374 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
3022, 24, 29ltled 11361 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
31 df-br 5149 . . . . . . . . 9 (((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ↔ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ≤ )
3230, 31sylib 217 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ≤ )
3322, 24opelxpd 5715 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ (ℝ × ℝ))
3432, 33elind 4194 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
35 ovollb2.2 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)
3634, 35fmptd 7113 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
37 eqid 2732 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
38 ovollb2.3 . . . . . . 7 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
3937, 38ovolsf 24988 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
4036, 39syl 17 . . . . 5 (𝜑𝑇:ℕ⟶(0[,)+∞))
4140frnd 6725 . . . 4 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
42 icossxr 13408 . . . 4 (0[,)+∞) ⊆ ℝ*
4341, 42sstrdi 3994 . . 3 (𝜑 → ran 𝑇 ⊆ ℝ*)
44 supxrcl 13293 . . 3 (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
4543, 44syl 17 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
46 ovollb2.7 . . . 4 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
4711rpred 13015 . . . 4 (𝜑𝐵 ∈ ℝ)
4846, 47readdcld 11242 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ)
4948rexrd 11263 . 2 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*)
50 2fveq3 6896 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (1st ‘(𝐹𝑛)) = (1st ‘(𝐹𝑚)))
51 oveq2 7416 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
5251oveq2d 7424 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((𝐵 / 2) / (2↑𝑛)) = ((𝐵 / 2) / (2↑𝑚)))
5350, 52oveq12d 7426 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
54 2fveq3 6896 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (2nd ‘(𝐹𝑛)) = (2nd ‘(𝐹𝑚)))
5554, 52oveq12d 7426 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
5653, 55opeq12d 4881 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
57 opex 5464 . . . . . . . . . . . . . . 15 ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩ ∈ V
5856, 35, 57fvmpt 6998 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
5958adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
6059fveq2d 6895 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
61 ovex 7441 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ V
62 ovex 7441 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) ∈ V
6361, 62op1st 7982 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))
6460, 63eqtrdi 2788 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
65 ovolfcl 24982 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
662, 65sylan 580 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
6766simp1d 1142 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
6812adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
69 nnnn0 12478 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7069adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ0)
71 nnexpcl 14039 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
7214, 70, 71sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℕ)
7372nnrpd 13013 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℝ+)
7468, 73rpdivcld 13032 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℝ+)
7567, 74ltsubrpd 13047 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) < (1st ‘(𝐹𝑚)))
7664, 75eqbrtrd 5170 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
7776adantlr 713 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
78 ovolfcl 24982 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
7936, 78sylan 580 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
8079simp1d 1142 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8180adantlr 713 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8267adantlr 713 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
835sselda 3982 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → 𝑧 ∈ ℝ)
8483adantr 481 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℝ)
85 ltletr 11305 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
8681, 82, 84, 85syl3anc 1371 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
8777, 86mpand 693 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ≤ 𝑧 → (1st ‘(𝐺𝑚)) < 𝑧))
8866simp2d 1143 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
8988, 74ltaddrpd 13048 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9059fveq2d 6895 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
9161, 62op2nd 7983 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))
9290, 91eqtrdi 2788 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9389, 92breqtrrd 5176 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9493adantlr 713 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9588adantlr 713 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
9679simp2d 1143 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
9796adantlr 713 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
98 lelttr 11303 . . . . . . . . . 10 ((𝑧 ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
9984, 95, 97, 98syl3anc 1371 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
10094, 99mpan2d 692 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 ≤ (2nd ‘(𝐹𝑚)) → 𝑧 < (2nd ‘(𝐺𝑚))))
10187, 100anim12d 609 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
102101reximdva 3168 . . . . . 6 ((𝜑𝑧𝐴) → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
103102ralimdva 3167 . . . . 5 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
104 ovolficc 24984 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
1055, 2, 104syl2anc 584 . . . . 5 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
106 ovolfioo 24983 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1075, 36, 106syl2anc 584 . . . . 5 (𝜑 → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
108103, 105, 1073imtr4d 293 . . . 4 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐺)))
1091, 108mpd 15 . . 3 (𝜑𝐴 ran ((,) ∘ 𝐺))
11038ovollb 24995 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐺)) → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11136, 109, 110syl2anc 584 . 2 (𝜑 → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11238fveq1i 6892 . . . . . . 7 (𝑇𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘)
113 fzfid 13937 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
114 rge0ssre 13432 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ
115 eqid 2732 . . . . . . . . . . . . . . 15 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
116115ovolfsf 24987 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
1172, 116syl 17 . . . . . . . . . . . . 13 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
118117adantr 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
119 elfznn 13529 . . . . . . . . . . . 12 (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
120 ffvelcdm 7083 . . . . . . . . . . . 12 ((((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
121118, 119, 120syl2an 596 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
122114, 121sselid 3980 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℝ)
123122recnd 11241 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℂ)
12411adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℝ+)
125124, 73rpdivcld 13032 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℝ+)
126125rpcnd 13017 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℂ)
127119, 126sylan2 593 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
128127adantlr 713 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
129113, 123, 128fsumadd 15685 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))))
13037ovolfsval 24986 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13136, 130sylan 580 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13288recnd 11241 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℂ)
13374rpcnd 13017 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℂ)
13467recnd 11241 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℂ)
135134, 133subcld 11570 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ ℂ)
136132, 133, 135addsubassd 11590 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
13792, 64oveq12d 7426 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = (((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))))
138132, 134, 126subadd23d 11592 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚)))))
139115ovolfsval 24986 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
1402, 139sylan 580 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
141140oveq1d 7423 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))))
142133, 134, 133subsub3d 11600 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))))
14368rpcnd 13017 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℂ)
14472nncnd 12227 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℂ)
14572nnne0d 12261 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ≠ 0)
146143, 143, 144, 145divdird 12027 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
147124rpcnd 13017 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℂ)
1481472halvesd 12457 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) + (𝐵 / 2)) = 𝐵)
149148oveq1d 7423 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (𝐵 / (2↑𝑚)))
150146, 149eqtr3d 2774 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) = (𝐵 / (2↑𝑚)))
151150oveq1d 7423 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
152142, 151eqtrd 2772 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
153152oveq2d 7424 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚)))))
154138, 141, 1533eqtr4d 2782 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
155136, 137, 1543eqtr4d 2782 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
156131, 155eqtrd 2772 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
157119, 156sylan2 593 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
158157adantlr 713 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
159 simpr 485 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
160 nnuz 12864 . . . . . . . . . 10 ℕ = (ℤ‘1)
161159, 160eleqtrdi 2843 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
162123, 128addcld 11232 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) ∈ ℂ)
163158, 161, 162fsumser 15675 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
164 eqidd 2733 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = (((abs ∘ − ) ∘ 𝐹)‘𝑚))
165164, 161, 123fsumser 15675 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
166 ovollb2.1 . . . . . . . . . . 11 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
167166fveq1i 6892 . . . . . . . . . 10 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
168165, 167eqtr4di 2790 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (𝑆𝑘))
16911adantr 481 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ+)
170169rpcnd 13017 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
171 geo2sum 15818 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
172159, 170, 171syl2anc 584 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
173168, 172oveq12d 7426 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
174129, 163, 1733eqtr3d 2780 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
175112, 174eqtrid 2784 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
176115, 166ovolsf 24988 . . . . . . . . . 10 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1772, 176syl 17 . . . . . . . . 9 (𝜑𝑆:ℕ⟶(0[,)+∞))
178177ffvelcdmda 7086 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ (0[,)+∞))
179114, 178sselid 3980 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ℝ)
180169rpred 13015 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ)
181 nnnn0 12478 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
182181adantl 482 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
183 nnexpcl 14039 . . . . . . . . . . . 12 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
18414, 182, 183sylancr 587 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℕ)
185184nnrpd 13013 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℝ+)
186169, 185rpdivcld 13032 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ+)
187186rpred 13015 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ)
188180, 187resubcld 11641 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ∈ ℝ)
18946adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
190177frnd 6725 . . . . . . . . . 10 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
191190, 42sstrdi 3994 . . . . . . . . 9 (𝜑 → ran 𝑆 ⊆ ℝ*)
192191adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
193177ffnd 6718 . . . . . . . . 9 (𝜑𝑆 Fn ℕ)
194 fnfvelrn 7082 . . . . . . . . 9 ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
195193, 194sylan 580 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
196 supxrub 13302 . . . . . . . 8 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑘) ∈ ran 𝑆) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
197192, 195, 196syl2anc 584 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
198180, 186ltsubrpd 13047 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) < 𝐵)
199188, 180, 198ltled 11361 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ≤ 𝐵)
200179, 188, 189, 180, 197, 199le2addd 11832 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
201175, 200eqbrtrd 5170 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
202201ralrimiva 3146 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
203 ffn 6717 . . . . 5 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
204 breq1 5151 . . . . . 6 (𝑦 = (𝑇𝑘) → (𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
205204ralrn 7089 . . . . 5 (𝑇 Fn ℕ → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
20640, 203, 2053syl 18 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
207202, 206mpbird 256 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
208 supxrleub 13304 . . . 4 ((ran 𝑇 ⊆ ℝ* ∧ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*) → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
20943, 49, 208syl2anc 584 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
210207, 209mpbird 256 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
2117, 45, 49, 111, 210xrletrd 13140 1 (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cin 3947  wss 3948  cop 4634   cuni 4908   class class class wbr 5148  cmpt 5231   × cxp 5674  ran crn 5677  ccom 5680   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  supcsup 9434  cc 11107  cr 11108  0cc0 11109  1c1 11110   + caddc 11112  +∞cpnf 11244  *cxr 11246   < clt 11247  cle 11248  cmin 11443   / cdiv 11870  cn 12211  2c2 12266  0cn0 12471  cuz 12821  +crp 12973  (,)cioo 13323  [,)cico 13325  [,]cicc 13326  ...cfz 13483  seqcseq 13965  cexp 14026  abscabs 15180  Σcsu 15631  vol*covol 24978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-oi 9504  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12974  df-ioo 13327  df-ico 13329  df-icc 13330  df-fz 13484  df-fzo 13627  df-seq 13966  df-exp 14027  df-hash 14290  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-clim 15431  df-sum 15632  df-ovol 24980
This theorem is referenced by:  ovollb2  25005
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