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Theorem ovolunlem1 23781
Description: Lemma for ovolun 23783. (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovolun.a (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
ovolun.b (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
ovolun.c (𝜑𝐶 ∈ ℝ+)
ovolun.s 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolun.t 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ovolun.u 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ovolun.f1 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
ovolun.f2 (𝜑𝐴 ran ((,) ∘ 𝐹))
ovolun.f3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
ovolun.g1 (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
ovolun.g2 (𝜑𝐵 ran ((,) ∘ 𝐺))
ovolun.g3 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
ovolun.h 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))
Assertion
Ref Expression
ovolunlem1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Distinct variable groups:   𝐶,𝑛   𝑛,𝐹   𝐴,𝑛   𝐵,𝑛   𝑛,𝐺   𝜑,𝑛
Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑈(𝑛)   𝐻(𝑛)

Proof of Theorem ovolunlem1
Dummy variables 𝑘 𝑧 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolun.a . . . . 5 (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
21simpld 495 . . . 4 (𝜑𝐴 ⊆ ℝ)
3 ovolun.b . . . . 5 (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
43simpld 495 . . . 4 (𝜑𝐵 ⊆ ℝ)
52, 4unssd 4085 . . 3 (𝜑 → (𝐴𝐵) ⊆ ℝ)
6 ovolun.g1 . . . . . . . . . . . 12 (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
7 elovolmlem 23758 . . . . . . . . . . . 12 (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
86, 7sylib 219 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
98adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
109ffvelrnda 6719 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
11 nneo 11916 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
1211adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
1312con2bid 356 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (((𝑛 + 1) / 2) ∈ ℕ ↔ ¬ (𝑛 / 2) ∈ ℕ))
1413biimpar 478 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((𝑛 + 1) / 2) ∈ ℕ)
15 ovolun.f1 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
16 elovolmlem 23758 . . . . . . . . . . . . 13 (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1715, 16sylib 219 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1817adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1918ffvelrnda 6719 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ((𝑛 + 1) / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2014, 19syldan 591 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2110, 20ifclda 4417 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) ∈ ( ≤ ∩ (ℝ × ℝ)))
22 ovolun.h . . . . . . . 8 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))
2321, 22fmptd 6744 . . . . . . 7 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24 eqid 2794 . . . . . . . 8 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
25 ovolun.u . . . . . . . 8 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
2624, 25ovolsf 23756 . . . . . . 7 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
2723, 26syl 17 . . . . . 6 (𝜑𝑈:ℕ⟶(0[,)+∞))
28 rge0ssre 12694 . . . . . 6 (0[,)+∞) ⊆ ℝ
29 fss 6398 . . . . . 6 ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
3027, 28, 29sylancl 586 . . . . 5 (𝜑𝑈:ℕ⟶ℝ)
3130frnd 6392 . . . 4 (𝜑 → ran 𝑈 ⊆ ℝ)
32 1nn 11499 . . . . . . 7 1 ∈ ℕ
33 1z 11862 . . . . . . . . . 10 1 ∈ ℤ
34 seqfn 13231 . . . . . . . . . 10 (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3533, 34mp1i 13 . . . . . . . . 9 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3625fneq1i 6323 . . . . . . . . . 10 (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
37 nnuz 12130 . . . . . . . . . . 11 ℕ = (ℤ‘1)
3837fneq2i 6324 . . . . . . . . . 10 (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3936, 38bitri 276 . . . . . . . . 9 (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
4035, 39sylibr 235 . . . . . . . 8 (𝜑𝑈 Fn ℕ)
41 fndm 6328 . . . . . . . 8 (𝑈 Fn ℕ → dom 𝑈 = ℕ)
4240, 41syl 17 . . . . . . 7 (𝜑 → dom 𝑈 = ℕ)
4332, 42syl5eleqr 2889 . . . . . 6 (𝜑 → 1 ∈ dom 𝑈)
4443ne0d 4223 . . . . 5 (𝜑 → dom 𝑈 ≠ ∅)
45 dm0rn0 5682 . . . . . 6 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
4645necon3bii 3035 . . . . 5 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
4744, 46sylib 219 . . . 4 (𝜑 → ran 𝑈 ≠ ∅)
481simprd 496 . . . . . . . 8 (𝜑 → (vol*‘𝐴) ∈ ℝ)
493simprd 496 . . . . . . . 8 (𝜑 → (vol*‘𝐵) ∈ ℝ)
5048, 49readdcld 10519 . . . . . . 7 (𝜑 → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
51 ovolun.c . . . . . . . 8 (𝜑𝐶 ∈ ℝ+)
5251rpred 12281 . . . . . . 7 (𝜑𝐶 ∈ ℝ)
5350, 52readdcld 10519 . . . . . 6 (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ)
54 ovolun.s . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
55 ovolun.t . . . . . . . . 9 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
56 ovolun.f2 . . . . . . . . 9 (𝜑𝐴 ran ((,) ∘ 𝐹))
57 ovolun.f3 . . . . . . . . 9 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
58 ovolun.g2 . . . . . . . . 9 (𝜑𝐵 ran ((,) ∘ 𝐺))
59 ovolun.g3 . . . . . . . . 9 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
601, 3, 51, 54, 55, 25, 15, 56, 57, 6, 58, 59, 22ovolunlem1a 23780 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
6160ralrimiva 3148 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
62 breq1 4967 . . . . . . . . 9 (𝑧 = (𝑈𝑘) → (𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6362ralrn 6722 . . . . . . . 8 (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6440, 63syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6561, 64mpbird 258 . . . . . 6 (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
66 brralrspcev 5024 . . . . . 6 (((((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘)
6753, 65, 66syl2anc 584 . . . . 5 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘)
68 ressxr 10534 . . . . . . 7 ℝ ⊆ ℝ*
6931, 68syl6ss 3903 . . . . . 6 (𝜑 → ran 𝑈 ⊆ ℝ*)
70 supxrbnd2 12565 . . . . . 6 (ran 𝑈 ⊆ ℝ* → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
7169, 70syl 17 . . . . 5 (𝜑 → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
7267, 71mpbid 233 . . . 4 (𝜑 → sup(ran 𝑈, ℝ*, < ) < +∞)
73 supxrbnd 12571 . . . 4 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ*, < ) < +∞) → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
7431, 47, 72, 73syl3anc 1364 . . 3 (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
75 nncn 11496 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
7675adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
77 1cnd 10485 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 1 ∈ ℂ)
78762timesd 11730 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
7978oveq1d 7034 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
8076, 76, 77, 79assraddsubd 10904 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
81 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
82 nnm1nn0 11788 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ0)
83 nnnn0addcl 11777 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ0) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
8481, 82, 83syl2anc2 585 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
8580, 84eqeltrd 2882 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
86 oveq1 7026 . . . . . . . . . . . . . . . 16 (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
8786eleq1d 2866 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
8886fveq2d 6545 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
89 oveq1 7026 . . . . . . . . . . . . . . . 16 (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
9089fvoveq1d 7041 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
9187, 88, 90ifbieq12d 4410 . . . . . . . . . . . . . 14 (𝑛 = ((2 · 𝑚) − 1) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
92 fvex 6554 . . . . . . . . . . . . . . 15 (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
93 fvex 6554 . . . . . . . . . . . . . . 15 (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
9492, 93ifex 4431 . . . . . . . . . . . . . 14 if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
9591, 22, 94fvmpt 6638 . . . . . . . . . . . . 13 (((2 · 𝑚) − 1) ∈ ℕ → (𝐻‘((2 · 𝑚) − 1)) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
9685, 95syl 17 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
97 2nn 11560 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℕ
98 nnmulcl 11511 . . . . . . . . . . . . . . . . . . . 20 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
9997, 81, 98sylancr 587 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
10099nncnd 11504 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
101 ax-1cn 10444 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
102 npcan 10745 . . . . . . . . . . . . . . . . . 18 (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
103100, 101, 102sylancl 586 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
104103oveq1d 7034 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
105 2cn 11562 . . . . . . . . . . . . . . . . . 18 2 ∈ ℂ
106 2ne0 11591 . . . . . . . . . . . . . . . . . 18 2 ≠ 0
107 divcan3 11174 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
108105, 106, 107mp3an23 1445 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
10976, 108syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
110104, 109eqtrd 2830 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
111110, 81eqeltrd 2882 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
112 nneo 11916 . . . . . . . . . . . . . . . 16 (((2 · 𝑚) − 1) ∈ ℕ → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
11385, 112syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
114113con2bid 356 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ ↔ ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
115111, 114mpbid 233 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
116115iffalsed 4394 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
117110fveq2d 6545 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹𝑚))
11896, 116, 1173eqtrd 2834 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚))
119 fveqeq2 6550 . . . . . . . . . . . 12 (𝑘 = ((2 · 𝑚) − 1) → ((𝐻𝑘) = (𝐹𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚)))
120119rspcev 3557 . . . . . . . . . . 11 ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚)) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚))
12185, 118, 120syl2anc 584 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚))
122 fveq2 6541 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐹𝑚) → (1st ‘(𝐻𝑘)) = (1st ‘(𝐹𝑚)))
123122breq1d 4974 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐹𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧 ↔ (1st ‘(𝐹𝑚)) < 𝑧))
124 fveq2 6541 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐹𝑚) → (2nd ‘(𝐻𝑘)) = (2nd ‘(𝐹𝑚)))
125124breq2d 4976 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐹𝑚) → (𝑧 < (2nd ‘(𝐻𝑘)) ↔ 𝑧 < (2nd ‘(𝐹𝑚))))
126123, 125anbi12d 630 . . . . . . . . . . . 12 ((𝐻𝑘) = (𝐹𝑚) → (((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘))) ↔ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
127126biimprcd 251 . . . . . . . . . . 11 (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ((𝐻𝑘) = (𝐹𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
128127reximdv 3235 . . . . . . . . . 10 (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → (∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
129121, 128syl5com 31 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
130129rexlimdva 3246 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
131130ralimdv 3144 . . . . . . 7 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
132 ovolfioo 23751 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
1332, 17, 132syl2anc 584 . . . . . . 7 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
134 ovolfioo 23751 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
1352, 23, 134syl2anc 584 . . . . . . 7 (𝜑 → (𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
136131, 133, 1353imtr4d 295 . . . . . 6 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐻)))
13756, 136mpd 15 . . . . 5 (𝜑𝐴 ran ((,) ∘ 𝐻))
138 oveq1 7026 . . . . . . . . . . . . . . . 16 (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
139138eleq1d 2866 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
140138fveq2d 6545 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
141 oveq1 7026 . . . . . . . . . . . . . . . 16 (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
142141fvoveq1d 7041 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
143139, 140, 142ifbieq12d 4410 . . . . . . . . . . . . . 14 (𝑛 = (2 · 𝑚) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
144 fvex 6554 . . . . . . . . . . . . . . 15 (𝐺‘((2 · 𝑚) / 2)) ∈ V
145 fvex 6554 . . . . . . . . . . . . . . 15 (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
146144, 145ifex 4431 . . . . . . . . . . . . . 14 if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
147143, 22, 146fvmpt 6638 . . . . . . . . . . . . 13 ((2 · 𝑚) ∈ ℕ → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
14899, 147syl 17 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
149109, 81eqeltrd 2882 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
150149iftrued 4391 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) = (𝐺‘((2 · 𝑚) / 2)))
151109fveq2d 6545 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺𝑚))
152148, 150, 1513eqtrd 2834 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺𝑚))
153 fveqeq2 6550 . . . . . . . . . . . 12 (𝑘 = (2 · 𝑚) → ((𝐻𝑘) = (𝐺𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺𝑚)))
154153rspcev 3557 . . . . . . . . . . 11 (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺𝑚)) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚))
15599, 152, 154syl2anc 584 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚))
156 fveq2 6541 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐺𝑚) → (1st ‘(𝐻𝑘)) = (1st ‘(𝐺𝑚)))
157156breq1d 4974 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐺𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧 ↔ (1st ‘(𝐺𝑚)) < 𝑧))
158 fveq2 6541 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐺𝑚) → (2nd ‘(𝐻𝑘)) = (2nd ‘(𝐺𝑚)))
159158breq2d 4976 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐺𝑚) → (𝑧 < (2nd ‘(𝐻𝑘)) ↔ 𝑧 < (2nd ‘(𝐺𝑚))))
160157, 159anbi12d 630 . . . . . . . . . . . 12 ((𝐻𝑘) = (𝐺𝑚) → (((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘))) ↔ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
161160biimprcd 251 . . . . . . . . . . 11 (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ((𝐻𝑘) = (𝐺𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
162161reximdv 3235 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → (∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
163155, 162syl5com 31 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
164163rexlimdva 3246 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
165164ralimdv 3144 . . . . . . 7 (𝜑 → (∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
166 ovolfioo 23751 . . . . . . . 8 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1674, 8, 166syl2anc 584 . . . . . . 7 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
168 ovolfioo 23751 . . . . . . . 8 ((𝐵 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
1694, 23, 168syl2anc 584 . . . . . . 7 (𝜑 → (𝐵 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
170165, 167, 1693imtr4d 295 . . . . . 6 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) → 𝐵 ran ((,) ∘ 𝐻)))
17158, 170mpd 15 . . . . 5 (𝜑𝐵 ran ((,) ∘ 𝐻))
172137, 171unssd 4085 . . . 4 (𝜑 → (𝐴𝐵) ⊆ ran ((,) ∘ 𝐻))
17325ovollb 23763 . . . 4 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴𝐵) ⊆ ran ((,) ∘ 𝐻)) → (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
17423, 172, 173syl2anc 584 . . 3 (𝜑 → (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
175 ovollecl 23767 . . 3 (((𝐴𝐵) ⊆ ℝ ∧ sup(ran 𝑈, ℝ*, < ) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < )) → (vol*‘(𝐴𝐵)) ∈ ℝ)
1765, 74, 174, 175syl3anc 1364 . 2 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ)
17753rexrd 10540 . . . 4 (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*)
178 supxrleub 12569 . . . 4 ((ran 𝑈 ⊆ ℝ* ∧ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*) → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
17969, 177, 178syl2anc 584 . . 3 (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
18065, 179mpbird 258 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
181176, 74, 53, 174, 180letrd 10646 1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1522  wcel 2080  wne 2983  wral 3104  wrex 3105  cun 3859  cin 3860  wss 3861  c0 4213  ifcif 4383   cuni 4747   class class class wbr 4964  cmpt 5043   × cxp 5444  dom cdm 5446  ran crn 5447  ccom 5450   Fn wfn 6223  wf 6224  cfv 6228  (class class class)co 7019  1st c1st 7546  2nd c2nd 7547  𝑚 cmap 8259  supcsup 8753  cc 10384  cr 10385  0cc0 10386  1c1 10387   + caddc 10389   · cmul 10391  +∞cpnf 10521  *cxr 10523   < clt 10524  cle 10525  cmin 10719   / cdiv 11147  cn 11488  2c2 11542  0cn0 11747  cz 11831  cuz 12093  +crp 12239  (,)cioo 12588  [,)cico 12590  seqcseq 13219  abscabs 14427  vol*covol 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463  ax-pre-sup 10464
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-1st 7548  df-2nd 7549  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-er 8142  df-map 8261  df-en 8361  df-dom 8362  df-sdom 8363  df-sup 8755  df-inf 8756  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-div 11148  df-nn 11489  df-2 11550  df-3 11551  df-n0 11748  df-z 11832  df-uz 12094  df-rp 12240  df-ioo 12592  df-ico 12594  df-fz 12743  df-fl 13012  df-seq 13220  df-exp 13280  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-ovol 23748
This theorem is referenced by:  ovolunlem2  23782
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