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Theorem ovollb2lem 25005
Description: Lemma for ovollb2 25006. (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 24986 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
42, 3syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
51, 4sstrd 3993 . . 3 (𝜑𝐴 ⊆ ℝ)
6 ovolcl 24995 . . 3 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
75, 6syl 17 . 2 (𝜑 → (vol*‘𝐴) ∈ ℝ*)
8 ovolfcl 24983 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
92, 8sylan 581 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
109simp1d 1143 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
11 ovollb2.6 . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ ℝ+)
1211rphalfcld 13028 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 / 2) ∈ ℝ+)
1312adantr 482 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
14 2nn 12285 . . . . . . . . . . . . . . 15 2 ∈ ℕ
15 nnnn0 12479 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1615adantl 483 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17 nnexpcl 14040 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1814, 16, 17sylancr 588 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℕ)
1918nnrpd 13014 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
2013, 19rpdivcld 13033 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ+)
2120rpred 13016 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ)
2210, 21resubcld 11642 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
239simp2d 1144 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2423, 21readdcld 11243 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
2510, 20ltsubrpd 13048 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < (1st ‘(𝐹𝑛)))
269simp3d 1145 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
2723, 20ltaddrpd 13049 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2810, 23, 24, 26, 27lelttrd 11372 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
2922, 10, 24, 25, 28lttrd 11375 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
3022, 24, 29ltled 11362 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
31 df-br 5150 . . . . . . . . 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 5716 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ (ℝ × ℝ))
3432, 33elind 4195 . . . . . . 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 7114 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
37 eqid 2733 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
38 ovollb2.3 . . . . . . 7 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
3937, 38ovolsf 24989 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
4036, 39syl 17 . . . . 5 (𝜑𝑇:ℕ⟶(0[,)+∞))
4140frnd 6726 . . . 4 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
42 icossxr 13409 . . . 4 (0[,)+∞) ⊆ ℝ*
4341, 42sstrdi 3995 . . 3 (𝜑 → ran 𝑇 ⊆ ℝ*)
44 supxrcl 13294 . . 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 13016 . . . 4 (𝜑𝐵 ∈ ℝ)
4846, 47readdcld 11243 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ)
4948rexrd 11264 . 2 (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*)
50 2fveq3 6897 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (1st ‘(𝐹𝑛)) = (1st ‘(𝐹𝑚)))
51 oveq2 7417 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
5251oveq2d 7425 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((𝐵 / 2) / (2↑𝑛)) = ((𝐵 / 2) / (2↑𝑚)))
5350, 52oveq12d 7427 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((1st ‘(𝐹𝑛)) − ((𝐵 / 2) / (2↑𝑛))) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
54 2fveq3 6897 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (2nd ‘(𝐹𝑛)) = (2nd ‘(𝐹𝑚)))
5554, 52oveq12d 7427 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → ((2nd ‘(𝐹𝑛)) + ((𝐵 / 2) / (2↑𝑛))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
5653, 55opeq12d 4882 . . . . . . . . . . . . . . 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 5465 . . . . . . . . . . . . . . 15 ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩ ∈ V
5856, 35, 57fvmpt 6999 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
5958adantl 483 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) = ⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
6059fveq2d 6896 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
61 ovex 7442 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ V
62 ovex 7442 . . . . . . . . . . . . 13 ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) ∈ V
6361, 62op1st 7983 . . . . . . . . . . . 12 (1st ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))
6460, 63eqtrdi 2789 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) = ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
65 ovolfcl 24983 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
662, 65sylan 581 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ≤ (2nd ‘(𝐹𝑚))))
6766simp1d 1143 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
6812adantr 482 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
69 nnnn0 12479 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
7069adantl 483 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ0)
71 nnexpcl 14040 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
7214, 70, 71sylancr 588 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℕ)
7372nnrpd 13014 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℝ+)
7468, 73rpdivcld 13033 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℝ+)
7567, 74ltsubrpd 13048 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) < (1st ‘(𝐹𝑚)))
7664, 75eqbrtrd 5171 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
7776adantlr 714 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)))
78 ovolfcl 24983 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
7936, 78sylan 581 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐺𝑚)) ≤ (2nd ‘(𝐺𝑚))))
8079simp1d 1143 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8180adantlr 714 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺𝑚)) ∈ ℝ)
8267adantlr 714 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℝ)
835sselda 3983 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → 𝑧 ∈ ℝ)
8483adantr 482 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℝ)
85 ltletr 11306 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) ∈ ℝ ∧ (1st ‘(𝐹𝑚)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
8681, 82, 84, 85syl3anc 1372 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < (1st ‘(𝐹𝑚)) ∧ (1st ‘(𝐹𝑚)) ≤ 𝑧) → (1st ‘(𝐺𝑚)) < 𝑧))
8777, 86mpand 694 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) ≤ 𝑧 → (1st ‘(𝐺𝑚)) < 𝑧))
8866simp2d 1144 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
8988, 74ltaddrpd 13049 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9059fveq2d 6896 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
9161, 62op2nd 7984 . . . . . . . . . . . 12 (2nd ‘⟨((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚)))
9290, 91eqtrdi 2789 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
9389, 92breqtrrd 5177 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9493adantlr 714 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚)))
9588adantlr 714 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℝ)
9679simp2d 1144 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
9796adantlr 714 . . . . . . . . . 10 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺𝑚)) ∈ ℝ)
98 lelttr 11304 . . . . . . . . . 10 ((𝑧 ∈ ℝ ∧ (2nd ‘(𝐹𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺𝑚)) ∈ ℝ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
9984, 95, 97, 98syl3anc 1372 . . . . . . . . 9 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → ((𝑧 ≤ (2nd ‘(𝐹𝑚)) ∧ (2nd ‘(𝐹𝑚)) < (2nd ‘(𝐺𝑚))) → 𝑧 < (2nd ‘(𝐺𝑚))))
10094, 99mpan2d 693 . . . . . . . 8 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 ≤ (2nd ‘(𝐹𝑚)) → 𝑧 < (2nd ‘(𝐺𝑚))))
10187, 100anim12d 610 . . . . . . 7 (((𝜑𝑧𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
102101reximdva 3169 . . . . . 6 ((𝜑𝑧𝐴) → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
103102ralimdva 3168 . . . . 5 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
104 ovolficc 24985 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
1055, 2, 104syl2anc 585 . . . . 5 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑚)))))
106 ovolfioo 24984 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1075, 36, 106syl2anc 585 . . . . 5 (𝜑 → (𝐴 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
108103, 105, 1073imtr4d 294 . . . 4 (𝜑 → (𝐴 ran ([,] ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐺)))
1091, 108mpd 15 . . 3 (𝜑𝐴 ran ((,) ∘ 𝐺))
11038ovollb 24996 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ((,) ∘ 𝐺)) → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11136, 109, 110syl2anc 585 . 2 (𝜑 → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
11238fveq1i 6893 . . . . . . 7 (𝑇𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘)
113 fzfid 13938 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
114 rge0ssre 13433 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ
115 eqid 2733 . . . . . . . . . . . . . . 15 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
116115ovolfsf 24988 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
1172, 116syl 17 . . . . . . . . . . . . 13 (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
118117adantr 482 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
119 elfznn 13530 . . . . . . . . . . . 12 (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
120 ffvelcdm 7084 . . . . . . . . . . . 12 ((((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
121118, 119, 120syl2an 597 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
122114, 121sselid 3981 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℝ)
123122recnd 11242 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℂ)
12411adantr 482 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℝ+)
125124, 73rpdivcld 13033 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℝ+)
126125rpcnd 13018 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℂ)
127119, 126sylan2 594 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
128127adantlr 714 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
129113, 123, 128fsumadd 15686 . . . . . . . 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 24987 . . . . . . . . . . . . 13 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13136, 130sylan 581 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))))
13288recnd 11242 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝐹𝑚)) ∈ ℂ)
13374rpcnd 13018 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℂ)
13467recnd 11242 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝐹𝑚)) ∈ ℂ)
135134, 133subcld 11571 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ ℂ)
136132, 133, 135addsubassd 11591 . . . . . . . . . . . . 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 7427 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = (((2nd ‘(𝐹𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))))
138132, 134, 126subadd23d 11593 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚)))))
139115ovolfsval 24987 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
1402, 139sylan 581 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))))
141140oveq1d 7424 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (((2nd ‘(𝐹𝑚)) − (1st ‘(𝐹𝑚))) + (𝐵 / (2↑𝑚))))
142133, 134, 133subsub3d 11601 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))))
14368rpcnd 13018 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℂ)
14472nncnd 12228 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℂ)
14572nnne0d 12262 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2↑𝑚) ≠ 0)
146143, 143, 144, 145divdird 12028 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
147124rpcnd 13018 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ ℕ) → 𝐵 ∈ ℂ)
1481472halvesd 12458 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → ((𝐵 / 2) + (𝐵 / 2)) = 𝐵)
149148oveq1d 7424 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (𝐵 / (2↑𝑚)))
150146, 149eqtr3d 2775 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) = (𝐵 / (2↑𝑚)))
151150oveq1d 7424 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹𝑚))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
152142, 151eqtrd 2773 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹𝑚))))
153152oveq2d 7425 . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
155136, 137, 1543eqtr4d 2783 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2nd ‘(𝐺𝑚)) − (1st ‘(𝐺𝑚))) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
156131, 155eqtrd 2773 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
157119, 156sylan2 594 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
158157adantlr 714 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
159 simpr 486 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
160 nnuz 12865 . . . . . . . . . 10 ℕ = (ℤ‘1)
161159, 160eleqtrdi 2844 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
162123, 128addcld 11233 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) ∈ ℂ)
163158, 161, 162fsumser 15676 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
164 eqidd 2734 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = (((abs ∘ − ) ∘ 𝐹)‘𝑚))
165164, 161, 123fsumser 15676 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
166 ovollb2.1 . . . . . . . . . . 11 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
167166fveq1i 6893 . . . . . . . . . 10 (𝑆𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
168165, 167eqtr4di 2791 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (𝑆𝑘))
16911adantr 482 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ+)
170169rpcnd 13018 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
171 geo2sum 15819 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
172159, 170, 171syl2anc 585 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
173168, 172oveq12d 7427 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
174129, 163, 1733eqtr3d 2781 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
175112, 174eqtrid 2785 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) = ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
176115, 166ovolsf 24989 . . . . . . . . . 10 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
1772, 176syl 17 . . . . . . . . 9 (𝜑𝑆:ℕ⟶(0[,)+∞))
178177ffvelcdmda 7087 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ (0[,)+∞))
179114, 178sselid 3981 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ℝ)
180169rpred 13016 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐵 ∈ ℝ)
181 nnnn0 12479 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
182181adantl 483 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
183 nnexpcl 14040 . . . . . . . . . . . 12 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
18414, 182, 183sylancr 588 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℕ)
185184nnrpd 13014 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℝ+)
186169, 185rpdivcld 13033 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ+)
187186rpred 13016 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ)
188180, 187resubcld 11642 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ∈ ℝ)
18946adantr 482 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
190177frnd 6726 . . . . . . . . . 10 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
191190, 42sstrdi 3995 . . . . . . . . 9 (𝜑 → ran 𝑆 ⊆ ℝ*)
192191adantr 482 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
193177ffnd 6719 . . . . . . . . 9 (𝜑𝑆 Fn ℕ)
194 fnfvelrn 7083 . . . . . . . . 9 ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
195193, 194sylan 581 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ran 𝑆)
196 supxrub 13303 . . . . . . . 8 ((ran 𝑆 ⊆ ℝ* ∧ (𝑆𝑘) ∈ ran 𝑆) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
197192, 195, 196syl2anc 585 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
198180, 186ltsubrpd 13048 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) < 𝐵)
199188, 180, 198ltled 11362 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ≤ 𝐵)
200179, 188, 189, 180, 197, 199le2addd 11833 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑆𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
201175, 200eqbrtrd 5171 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
202201ralrimiva 3147 . . . 4 (𝜑 → ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
203 ffn 6718 . . . . 5 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
204 breq1 5152 . . . . . 6 (𝑦 = (𝑇𝑘) → (𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ (𝑇𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
205204ralrn 7090 . . . . 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 257 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
208 supxrleub 13305 . . . 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 585 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
210207, 209mpbird 257 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
2117, 45, 49, 111, 210xrletrd 13141 1 (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cin 3948  wss 3949  cop 4635   cuni 4909   class class class wbr 5149  cmpt 5232   × cxp 5675  ran crn 5678  ccom 5681   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  supcsup 9435  cc 11108  cr 11109  0cc0 11110  1c1 11111   + caddc 11113  +∞cpnf 11245  *cxr 11247   < clt 11248  cle 11249  cmin 11444   / cdiv 11871  cn 12212  2c2 12267  0cn0 12472  cuz 12822  +crp 12974  (,)cioo 13324  [,)cico 13326  [,]cicc 13327  ...cfz 13484  seqcseq 13966  cexp 14027  abscabs 15181  Σcsu 15632  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-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-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-icc 13331  df-fz 13485  df-fzo 13628  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-sum 15633  df-ovol 24981
This theorem is referenced by:  ovollb2  25006
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