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Theorem ovolunlem1 24861
Description: Lemma for ovolun 24863. (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 4146 . . 3 (𝜑 → (𝐴𝐵) ⊆ ℝ)
6 ovolun.g1 . . . . . . . . . . . 12 (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
7 elovolmlem 24838 . . . . . . . . . . . 12 (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
86, 7sylib 217 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
98adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
109ffvelcdmda 7035 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
11 nneo 12587 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
1211adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
1312con2bid 354 . . . . . . . . . . 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 24838 . . . . . . . . . . . . 13 (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1715, 16sylib 217 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1817adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1918ffvelcdmda 7035 . . . . . . . . . 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 4521 . . . . . . . 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 7062 . . . . . . 7 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24 eqid 2736 . . . . . . . 8 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
25 ovolun.u . . . . . . . 8 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
2624, 25ovolsf 24836 . . . . . . 7 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
2723, 26syl 17 . . . . . 6 (𝜑𝑈:ℕ⟶(0[,)+∞))
28 rge0ssre 13373 . . . . . 6 (0[,)+∞) ⊆ ℝ
29 fss 6685 . . . . . 6 ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
3027, 28, 29sylancl 586 . . . . 5 (𝜑𝑈:ℕ⟶ℝ)
3130frnd 6676 . . . 4 (𝜑 → ran 𝑈 ⊆ ℝ)
32 1nn 12164 . . . . . . 7 1 ∈ ℕ
33 1z 12533 . . . . . . . . . 10 1 ∈ ℤ
34 seqfn 13918 . . . . . . . . . 10 (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3533, 34mp1i 13 . . . . . . . . 9 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3625fneq1i 6599 . . . . . . . . . 10 (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
37 nnuz 12806 . . . . . . . . . . 11 ℕ = (ℤ‘1)
3837fneq2i 6600 . . . . . . . . . 10 (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3936, 38bitri 274 . . . . . . . . 9 (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
4035, 39sylibr 233 . . . . . . . 8 (𝜑𝑈 Fn ℕ)
4140fndmd 6607 . . . . . . 7 (𝜑 → dom 𝑈 = ℕ)
4232, 41eleqtrrid 2845 . . . . . 6 (𝜑 → 1 ∈ dom 𝑈)
4342ne0d 4295 . . . . 5 (𝜑 → dom 𝑈 ≠ ∅)
44 dm0rn0 5880 . . . . . 6 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
4544necon3bii 2996 . . . . 5 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
4643, 45sylib 217 . . . 4 (𝜑 → ran 𝑈 ≠ ∅)
471simprd 496 . . . . . . . 8 (𝜑 → (vol*‘𝐴) ∈ ℝ)
483simprd 496 . . . . . . . 8 (𝜑 → (vol*‘𝐵) ∈ ℝ)
4947, 48readdcld 11184 . . . . . . 7 (𝜑 → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
50 ovolun.c . . . . . . . 8 (𝜑𝐶 ∈ ℝ+)
5150rpred 12957 . . . . . . 7 (𝜑𝐶 ∈ ℝ)
5249, 51readdcld 11184 . . . . . 6 (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ)
53 ovolun.s . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
54 ovolun.t . . . . . . . . 9 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
55 ovolun.f2 . . . . . . . . 9 (𝜑𝐴 ran ((,) ∘ 𝐹))
56 ovolun.f3 . . . . . . . . 9 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
57 ovolun.g2 . . . . . . . . 9 (𝜑𝐵 ran ((,) ∘ 𝐺))
58 ovolun.g3 . . . . . . . . 9 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
591, 3, 50, 53, 54, 25, 15, 55, 56, 6, 57, 58, 22ovolunlem1a 24860 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
6059ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
61 breq1 5108 . . . . . . . . 9 (𝑧 = (𝑈𝑘) → (𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6261ralrn 7038 . . . . . . . 8 (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6340, 62syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6460, 63mpbird 256 . . . . . 6 (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
65 brralrspcev 5165 . . . . . 6 (((((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘)
6652, 64, 65syl2anc 584 . . . . 5 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘)
67 ressxr 11199 . . . . . . 7 ℝ ⊆ ℝ*
6831, 67sstrdi 3956 . . . . . 6 (𝜑 → ran 𝑈 ⊆ ℝ*)
69 supxrbnd2 13241 . . . . . 6 (ran 𝑈 ⊆ ℝ* → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
7068, 69syl 17 . . . . 5 (𝜑 → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
7166, 70mpbid 231 . . . 4 (𝜑 → sup(ran 𝑈, ℝ*, < ) < +∞)
72 supxrbnd 13247 . . . 4 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ*, < ) < +∞) → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
7331, 46, 71, 72syl3anc 1371 . . 3 (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
74 nncn 12161 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
7574adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
76 1cnd 11150 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 1 ∈ ℂ)
77752timesd 12396 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
7877oveq1d 7372 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
7975, 75, 76, 78assraddsubd 11569 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
80 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
81 nnm1nn0 12454 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ0)
82 nnnn0addcl 12443 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ0) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
8380, 81, 82syl2anc2 585 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
8479, 83eqeltrd 2838 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
85 oveq1 7364 . . . . . . . . . . . . . . . 16 (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
8685eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
8785fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
88 oveq1 7364 . . . . . . . . . . . . . . . 16 (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
8988fvoveq1d 7379 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
9086, 87, 89ifbieq12d 4514 . . . . . . . . . . . . . 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))))
91 fvex 6855 . . . . . . . . . . . . . . 15 (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
92 fvex 6855 . . . . . . . . . . . . . . 15 (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
9391, 92ifex 4536 . . . . . . . . . . . . . 14 if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
9490, 22, 93fvmpt 6948 . . . . . . . . . . . . 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))))
9584, 94syl 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))))
96 2nn 12226 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℕ
97 nnmulcl 12177 . . . . . . . . . . . . . . . . . . . 20 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
9896, 80, 97sylancr 587 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
9998nncnd 12169 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
100 ax-1cn 11109 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
101 npcan 11410 . . . . . . . . . . . . . . . . . 18 (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
10299, 100, 101sylancl 586 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
103102oveq1d 7372 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
104 2cn 12228 . . . . . . . . . . . . . . . . . 18 2 ∈ ℂ
105 2ne0 12257 . . . . . . . . . . . . . . . . . 18 2 ≠ 0
106 divcan3 11839 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
107104, 105, 106mp3an23 1453 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
10875, 107syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
109103, 108eqtrd 2776 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
110109, 80eqeltrd 2838 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
111 nneo 12587 . . . . . . . . . . . . . . . 16 (((2 · 𝑚) − 1) ∈ ℕ → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
11284, 111syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
113112con2bid 354 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ ↔ ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
114110, 113mpbid 231 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
115114iffalsed 4497 . . . . . . . . . . . 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)))
116109fveq2d 6846 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹𝑚))
11795, 115, 1163eqtrd 2780 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚))
118 fveqeq2 6851 . . . . . . . . . . . 12 (𝑘 = ((2 · 𝑚) − 1) → ((𝐻𝑘) = (𝐹𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚)))
119118rspcev 3581 . . . . . . . . . . 11 ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚)) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚))
12084, 117, 119syl2anc 584 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚))
121 fveq2 6842 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐹𝑚) → (1st ‘(𝐻𝑘)) = (1st ‘(𝐹𝑚)))
122121breq1d 5115 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐹𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧 ↔ (1st ‘(𝐹𝑚)) < 𝑧))
123 fveq2 6842 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐹𝑚) → (2nd ‘(𝐻𝑘)) = (2nd ‘(𝐹𝑚)))
124123breq2d 5117 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐹𝑚) → (𝑧 < (2nd ‘(𝐻𝑘)) ↔ 𝑧 < (2nd ‘(𝐹𝑚))))
125122, 124anbi12d 631 . . . . . . . . . . . 12 ((𝐻𝑘) = (𝐹𝑚) → (((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘))) ↔ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
126125biimprcd 249 . . . . . . . . . . 11 (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ((𝐻𝑘) = (𝐹𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
127126reximdv 3167 . . . . . . . . . 10 (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → (∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
128120, 127syl5com 31 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
129128rexlimdva 3152 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
130129ralimdv 3166 . . . . . . 7 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
131 ovolfioo 24831 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
1322, 17, 131syl2anc 584 . . . . . . 7 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
133 ovolfioo 24831 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
1342, 23, 133syl2anc 584 . . . . . . 7 (𝜑 → (𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
135130, 132, 1343imtr4d 293 . . . . . 6 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐻)))
13655, 135mpd 15 . . . . 5 (𝜑𝐴 ran ((,) ∘ 𝐻))
137 oveq1 7364 . . . . . . . . . . . . . . . 16 (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
138137eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
139137fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
140 oveq1 7364 . . . . . . . . . . . . . . . 16 (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
141140fvoveq1d 7379 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
142138, 139, 141ifbieq12d 4514 . . . . . . . . . . . . . 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))))
143 fvex 6855 . . . . . . . . . . . . . . 15 (𝐺‘((2 · 𝑚) / 2)) ∈ V
144 fvex 6855 . . . . . . . . . . . . . . 15 (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
145143, 144ifex 4536 . . . . . . . . . . . . . 14 if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
146142, 22, 145fvmpt 6948 . . . . . . . . . . . . 13 ((2 · 𝑚) ∈ ℕ → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
14798, 146syl 17 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
148108, 80eqeltrd 2838 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
149148iftrued 4494 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) = (𝐺‘((2 · 𝑚) / 2)))
150108fveq2d 6846 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺𝑚))
151147, 149, 1503eqtrd 2780 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺𝑚))
152 fveqeq2 6851 . . . . . . . . . . . 12 (𝑘 = (2 · 𝑚) → ((𝐻𝑘) = (𝐺𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺𝑚)))
153152rspcev 3581 . . . . . . . . . . 11 (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺𝑚)) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚))
15498, 151, 153syl2anc 584 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚))
155 fveq2 6842 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐺𝑚) → (1st ‘(𝐻𝑘)) = (1st ‘(𝐺𝑚)))
156155breq1d 5115 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐺𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧 ↔ (1st ‘(𝐺𝑚)) < 𝑧))
157 fveq2 6842 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐺𝑚) → (2nd ‘(𝐻𝑘)) = (2nd ‘(𝐺𝑚)))
158157breq2d 5117 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐺𝑚) → (𝑧 < (2nd ‘(𝐻𝑘)) ↔ 𝑧 < (2nd ‘(𝐺𝑚))))
159156, 158anbi12d 631 . . . . . . . . . . . 12 ((𝐻𝑘) = (𝐺𝑚) → (((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘))) ↔ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
160159biimprcd 249 . . . . . . . . . . 11 (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ((𝐻𝑘) = (𝐺𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
161160reximdv 3167 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → (∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
162154, 161syl5com 31 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
163162rexlimdva 3152 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
164163ralimdv 3166 . . . . . . 7 (𝜑 → (∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
165 ovolfioo 24831 . . . . . . . 8 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1664, 8, 165syl2anc 584 . . . . . . 7 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
167 ovolfioo 24831 . . . . . . . 8 ((𝐵 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
1684, 23, 167syl2anc 584 . . . . . . 7 (𝜑 → (𝐵 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
169164, 166, 1683imtr4d 293 . . . . . 6 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) → 𝐵 ran ((,) ∘ 𝐻)))
17057, 169mpd 15 . . . . 5 (𝜑𝐵 ran ((,) ∘ 𝐻))
171136, 170unssd 4146 . . . 4 (𝜑 → (𝐴𝐵) ⊆ ran ((,) ∘ 𝐻))
17225ovollb 24843 . . . 4 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴𝐵) ⊆ ran ((,) ∘ 𝐻)) → (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
17323, 171, 172syl2anc 584 . . 3 (𝜑 → (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
174 ovollecl 24847 . . 3 (((𝐴𝐵) ⊆ ℝ ∧ sup(ran 𝑈, ℝ*, < ) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < )) → (vol*‘(𝐴𝐵)) ∈ ℝ)
1755, 73, 173, 174syl3anc 1371 . 2 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ)
17652rexrd 11205 . . . 4 (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*)
177 supxrleub 13245 . . . 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)))
17868, 176, 177syl2anc 584 . . 3 (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
17964, 178mpbird 256 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
180175, 73, 52, 173, 179letrd 11312 1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  cun 3908  cin 3909  wss 3910  c0 4282  ifcif 4486   cuni 4865   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  +crp 12915  (,)cioo 13264  [,)cico 13266  seqcseq 13906  abscabs 15119  vol*covol 24826
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ioo 13268  df-ico 13270  df-fz 13425  df-fl 13697  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-ovol 24828
This theorem is referenced by:  ovolunlem2  24862
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